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Craig,  of  Johns  Hopkins  University. 


HIGH  MASONRY 


DAMS. 


JOHN  B.  McMASTEE,  0.  E. 

AUTHOR  OF  "BRIDGE  AND  TUNNEL  CENTRES.' 


NEW   YORK: 

D.  VAN  NOSTRAND,  PUBLISHER, 
23  MURRAY  AND  27  WARREN  STREET. 

1876. 


PREFACE. 


In  the  preparation  of  the  following  treatise 
three  points  have  been  constantly  in  view,  to 
avoid  as  far  as  possible  all  purely  theoretical 
discussion,  to  discover  the  most  economical 
forms  of  profiles  consistent  with  perfect 
strength,  and  to  consider  none  that  have  not, 
after  repeated  practical  application  demon 
strated  their  excellence,  even  under  the  severest 
tests.  To  treat  the  subject,  however,  in  a  logical 
way,  it  has  been  found  best  to  begin  with  a 
theoretical  determination  of  the  strongest  and 
at  the  same  time  least  expensive  form  of  profile, 
and  afterwards  to  modify  this  to  meet  the  re 
quirements  that  arise  in  actual  construction. 

The  theoretical  type  is,  as  I  have  attempted 
to  show,  that  composed  of  a  vertical  face  on 
the  inner,  and  a  concave  surface  on  the  outer 
side.  Of  this,  there  are,  of  course,  an  almost 
unlimited  number  of  possible  modifications. 
But  when  we  impose  the  condition  of  economy, 
the  number  of  really  useful  ones  dwindle  down 
to  less  than  half  a  dozen.  Those  treated  of  in 
the  present  work  number  four.  The  first  (il 
lustrated  in  Fig.  9),  is,  beyond  all  doubt,  the 


very  best.  It  has  indeed,  been  often  urged 
against  this  type  of  profile,  that  it  is  difficult 
to  determine  with  accuracy  the  equations  of 
the  logarithmic  curves  forming  the  bounding 
faces,  as  also  to  cut  the  facing  stones  to  such 
a  curve.  As  to  the  first  objection,  no  equations 
can  surely  be  simpler  than  those  we  have 
given,  while  the  second  is  a  difficulty  most 
easily  removed,  not  by  argument,  but  by  de 
termination. 

The  three  other  types  are  also  profiles  of 
equal  resistance,  and  are  treated  of  so  fully  in 
the  work  as  to  call  for  no  remark  here.  It  will 
also  be  observed  that  I  have  touched  very 
lightly  on  the  sliding  of  dams  on  their  founda 
tion,  or  of  any  portion  of  them  along  a  hori 
zontal  joint.  This  has  been  done,  because, 
though  I  have  examined  as  fully  as  possible  the 
causes  that  have  led  to  the  destruction  of  dams 
of  all  style  of  profile  and  of  all  heights,  both 
abroad  and  in  this  country,  I  have  been  able 
to  find  extremely  few  that  may  justly  be  said 
to  have  yielded  by  sliding.  It  has  almost  in 
variably  been  by  revolving  about  an  axis  near 
the  outer  face,  caused  by  taking  too  great  a 
limit  of  vertical  pressure,  and  thus  throwing 
the  line  of  resistance,  when  full,  too  far  out 
ward  from  the  centre  of  thickness. 

JOHN  B.  McMASTER. 
YORK,  February,  1876. 


HIGH  MASONEY  DAMS, 


THE  subject  proposed  for  consideration 
in  the  following  work  is  that  of  the  pro 
file  of  masonry  dams  of  such  height, 
breadth  and  general  dimensions  as  would 
be  required  for  reservoir  purposes,  or  for 
impounding  the  waters  of  rivers  and 
large  streams  for  mill  or  irrigation  use. 
We  would  observe,  however,  at  the  out 
set,  that  as  this  matter  has  already  been 
treated  with  such  fullness  by  several 
writers,  and  especially  by  MM.  Delocre 
and  Sazilly — to  whose  excellent  "me 
moirs  "  we  are  greatly  indebted — we  can 
hope  to  add  little  that  is  really  new,  but 
shall  endeavor,  by  drawing  from  many 
sources,  to  supply  our  own  deficiency,  to 
diminish  the  errors  of  others,  and  thus 
obtain  results  very  much  more  accurate 


6 

than  could  be  derived  if  we  relied  solely 
on  ourselves. 

Before,  however,  we  take  up  the  con 
sideration  of  the  matter  of  the  form  of 
profile  that  shall  combine  the  greatest 
strength  with  the  least  amount  of  mate 
rial,  there  are  a  number  of  important 
points  to  be  considered  somewhat  in  de 
tail.  Thus,  it  is  necessary,  in  the  first 
place,  that  we  should  know  the  forces  to 
which  dams  are  subjected,  their  kind? 
whether  constant  or  variable,  the  meth 
ods  of  determining  their  direction  and 
calculating  their  intensity,  and  the  ef 
fects  they  are  likely  to  produce,  and 
these  matters  being  known,  we  may  pass 
to  the  consideration  of  the  conditions  of 
stability,  first  when  the  dam  has  only  its 
own  weight  to  support,  and,  secondly, 
when  it  has  to  withstand  both  its  own 
weight  and  the  pressure  of  the  water. 
We  may  then  deduce  a  theoretical  profile 
of  equal  resistance,  and,  finally,  adopt 
one  so  modified  by  the  requirements  of 
practice  and  suggestion  of  experience, 
that  it  shall  serve  as  a  profile  type,  ful- 


filling  to  the  utmost  the  requirements  of 
great  strength  and  stability,  beauty  of 
outline  and  economy  of  material. 

Now,  it  becomes  evident,  after  a  mo 
ment's  consideration,  that  there  are  but 
two  forces  that  may  at  any  time  be  re 
garded  as  acting  with  vigor  on  a  dam, 
and  these  are,  the  weight  of  the  mason 
ry,  cement  and  other  material  composing 
the  structure,  and  the  pressure  or  thrust 
of  the  water  whose  flow  it  checks.  The 
first  becomes,  to  all  intents  and  purposes, 
a  constant  quantity  as  soon  as  the  dam 
is  finished,  and  continues  so  for  ever 
after,  acting  vertically  downwards 
through  the  centre  of  gravity  of  the 
mass.  But,  on  the  other  hand,  the  latter 
force  is  one  of  great  variability.  For, 
as  its  intensity  at  any  moment  depends 
on  the  depth  or  head  of  water  behind 
the  dam,  increasing  as  the  water  deepens 
and1  decreasing  as  the  water  falls,  and 
the  head  of  water,  especially  in  reservoirs 
used  for  mill  or  irrigation  purposes,  be 
ing  subject  to  frequent  rise  and  fall,  it 
follows  that  this  thrust  must  be  consid- 


8 

ered  as  a  variable  quantity  and  treated 
accordingly.  It  is,  moreover,  to  be  ob 
served,  that  this  thrust  acts  horizontally, 
and  unlike  the  weight,  is  not  distributed 
uniformly  over  the  entire  face  of  the  dam, 
being  almost,  if  not  quite,  zero  at  the 
point  where  the  water  cuts  the  masonry, 
and  growing  greater  and  greater  as  we 
descend  towards  the  foot  of  the  dam. 
The  weight,  it  is  true,  also  increases  as 
we  go  from  the  top  to  the  bottom,  yet, 
if  we  suppose  the  dam  to  be  at  any  point . 
ten  feet  thick,  the  pressure  on  any  hori 
zontal  section  taken  at  that  point  will  be 
everywhere  the  same,  and  this  is  by  no 
means  the  case  if  we  take  an  area  ten 
feet  square  on  the  water  face  of  the 
dam,  and  against  which  the  fluid  presses. 
In  order  that  the  dam  may  not  yield 
under  the  first  force,  and  be  thrown 
down  by  the  greatness  of  its  own  weight, 
it  is  necessary,  should  the  structure  be 
of  such  height,  or  the  material  of  such 
heaviness,  that  the  pressure  per  unit  of 
surface  at  any  horizontal  section  is  in 
excess  of  the  "limit  of  pressure"  for 


masonry,  that  the  surface  of  the  section 
be  increased  so  that  the  pressure  being 
distributed  over  a  more  extended  area 
the  load  at  each  unit  of  surface  shall  be 
less.  The  second  force,  or  thrust  of  the 
water,  is  resisted  at  any  point  by  the 
weight  of  the  masonry  above  that  point, 
and  by  the  friction  of  the  stones,  which 
is  of  course  dependent  on  the  weight. 
Some  resistance  is  indeed  afforded  by  the 
bonding  power  of  the  hydraulic  mortar 
used  in  setting  the  stones,  but  this  is  so 
^small  that  precautions  of  safety  require 
that  it  shall  in  all  calculations  be  disre 
garded  entirely. 

But  these  two  forces,  the  weight  act 
ing  vertically  downwards  and  the  thrust 
of  the  water  acting  horizontally,  counter 
act  each  other  to  a  certain  extent,  and 
give  rise  to  a  third  power  or  resultant, 
the  position  of  which,  as  regards  the  base, 
will  determine  the  stability  of  the  dam. 
To  illustrate,  let  A  B  C  D  (Fig.  1)  repre 
sent  the  profile  of  a  dam  composed  of 
horizontal  courses  of  masonry  bedded 
on  each  other,  and  K  the  centre  of  gravi- 


10 


FIG,  I. 


ty  of  the  mass,  lying  above  the  line  E  F. 
Represent  by  K  G  the  direction  and  in 
tensity  of  the  weight  of  AF,  and  by 
KP  the  direction  and  intensity  of  the 
thrust  of  the  water  from  D  to  F.  Then, 
constructing  in  the  usual  way  the  paral 
lelogram  P  K  Gr  R,  we  shall  have  for  the 
resultant  of  KP  and  KG,  the  line  KR. 
Now,  supposing  the  dam  to  be  perfectly 
secure  as  to  its  weight,  the  force  P  of  the 
water  can  demolish  the  wall  only,  when, 
exceeding  the  weight  and  friction  K  G, 
it  shoves  the  mass  AF  along  the  joint 
E  F,  or  causes  it  to  rotate  about  an  axis 
through  E.  Which  of  these  motions, 


11 

the  slipping  or  rotating,  shall  take  place 
depends  entirely  on  the  magnitude  and 
direction  of  the  resultant  K  R.  If  the 
pressure  of  the  water  is  so  large  com 
pared  with  the  weight  that  the  angle 
RK  G,  which  the  resultant  makes  with 
the  vertical,  is  larger  than  the  angle  of 
friction  (32°  for  masonry  on  masonry), 
the  mass  A  F  will  then  slide  along  the 
line  EF;  while  if  the  position  of  the  re 
sultant  is  such  that  it  passes  without  the 
base  B  C,  then  rotation  will  take  place 
about  the  axis  of  E.  Of  these  two  mo 
tions,  the  latter  is  in  practice  the  most 
likely  to  occur,  inasmuch  as  in  nine  cases 
out  of  ten  when  rotation  does  take  place 
it  does  so  about  some  point  as  E',  nearer 
the  resultant  than  E,  because  the  press 
ure  concentrated  at  E,  breaks  off  the 
stone,  and  thus  throws  the  axis  of  rota 
tion  nearer  the  resultant. 

The  condition  of  stability  then,  in  dams 
that  do  not  transmit  laterally  to  the 
sides  of  the  valley,  the  pressure  they  sus 
tain  (and  this  is  the  ease  in  all  large 
dams)  is,  that  they  must  resist  this  press- 


12 

ure  at  every  point  by  their  own  weight. 
If  the  material  employed  were  of  con 
siderable  resisting  power,  as  well  as  the 
soil  of  the  foundation,  and  if  there  were 
between  them  an  unlimited  degree  of 
adhesion,  the  only  condition  of  stability 
to  be  fulfilled  would  be,  as  we  have  just 
seen,  to  give  the  wall  such  a  profile  that 
the  resultant  of  the  thrust  of  the  water 
and  the  weight  of  the  dam  shall  pass 
within  the  polygon  of  the  base.  But 
this  condition  is  not  found  sufficient  in 
practice  ;  the  material  and  the  soil  of 
the  foundation  will,  in  fact,  support  only 
a  limited  pressure  (depending  on  their 
nature),  and  they  have  not  between 
them  an  unlimited  degree  of  adhesion. 
Hence,  the  two  following  indispensable 
conditions  : 

1°  The  several  courses  of  masonry  in 
the  wall  must  be  incapable  of  slipping 
the  one  over  the  other,  and  the  wall  in 
capable  of  sliding  on  its  base. 

2°  In  no  point  of  the  structure  may 
the  material  employed,  or  the  soil  of  the 
foundation  be  required  to  bear  too  great 


13 


a  pressure.     To  begin  with  the  first  con 
dition. 

STABILITY   AS   TO    SLIPPING. 

We  shall  take  up  first  the  condition 
of  stability  as  to  the  slipping  of  the  va 
rious  courses  of  masonry,  and  then  pass 
to  that  of  the  entire  dam.  The  first 
thing  to  be  now  determined,  is  the  hori 
zontal  thrust  of  the  water.  Suppose 
A  B  C  D  (Fig.  2)  to  represent  the  face  of 
a  dam  pressed  by  water,  and  let  h=A  J 
denote  the  height;  a=  J  C  the  projection 
of  the  slope  of  the  dam  on  the  horizon 
tal  plane;  and,  finally,  let  1=  A  B  denote 
the  length  of  the  dam,  and  b=  A  G  is 
breadth  across  the  top.  Then  will  the 
vertical  pressure  of  the  water  on  the  face 
A  B  C  D  be  expressed  by 

al-  y—\  alliy    ...     1 

and  the  horizontal  thrust  by  the*  expres 
sion 


14 


in  each  of  which  y  denotes  the  density 
of  the  water.  These  equations  are  ob 
tained  as  follows  : 

Let  E  P,  in  Fig.  2,  represent  the  nor 
mal  pressure  of  the  water  on  the  surface 
A  C,  which  we  will  call  F,  and  resolve  it 


15 

into  two  components,  one  vertical  E  P', 
and  one  horizontal  E  P",  and  call  them 
respectively  P'  and  P".  Then  expressing 
the  angle  P  E  P"  made  by  the  horizontal 
component  P"  and  the  normal  E  P,  by  a 

we  shall  have  from  the  triangle  E  P  P" 
pp// 

-^5  =  sin  P  E  P"  or  sin  a. 
±L  Jr 

But  PP"=EP'=P',  hence 

P'  ") 

p-=sin  a     or  P'  =  P  sin  a. 

In  the  same  way  we  find 
p// 

—  =  cos  a     or  P"=:P  cos  a. 

Now,  let  a  projection  A'B'  C  D,  of  the 
surface  A  B  0  D,  be  made  on  a  plane  at 
right  angles  to  P",  and  call  the  area  of 
the  projected  surface  F'.  Then  will  F' 
=  F  cos  AC  A',  or  since  the  angle  of 
inclination  A  C  A'  of  the  surface  to  its 
projection  is  equal  to  the  angle  P  E  P" 
=  a,  between  the  normal  to  A  C,  and 
the  perpendicular  to  A7  C,  we  shall 

F' 
have  F'=F  cos  a  or  cos  a=^r«    But  cos 

JD        TD// 

a  is  by  equation  3  equal  to  ^-,  and 
therefore, 


16 
"p//      TJ^'  TT' 

From  the  principles  of  mechanics,  we 
know  that  the  pressure  P  of  water  on 
any  given  area  is  the  product  of  the 
area,^the  height  h  of  the  water,  and  its 
density  y,  so  that  in  the  present  instance 
F  being  the  area  of  the  surface  A  B  C  D, 
we  shall  have  for  the  value  of  P  the  ex 
pression  P=F  h  y,  and  this  substituted 
in  equation  4  gives 

P"=FAy  —  or  F'  hy  .     .     .     5. 

Therefore  is  the  pressure  with  which 
water  presses  against  a  surface  in  a  given 
direction  equal  to  the  weight  of  a  column 
of  water,  which  has  for  its  base  the  pro 
jection  of  the  surface  pressed,  and  for 
height  the  depth  of  the  centre  of  gravi 
ty  of  the  surface  below  the  top  of  the 
water.  We  see,  moreover,  from  the 
above,  that  since  the  projection  at  right 
angles  to  the  vertical  is  the  horizontal, 
and  the  projection  at  right  angles  to  the 
horizontal  is  the  vertical  projection,  the 


11 

vertical  component  of  the  pressure  of 
water  against  a  surface  may  be  found  if 
the  horizontal  projection,  or  its  trace,  be 
considered  as  the  surface  pressed,  and, 
on  the  other  hand,  the  horizontal  com 
ponent  may  be  found  if  the  vertical  pro 
jection  of  the  surface,  or  its  trace,  be 
considered  as  the  surface  pressed. 

Applying  these  two  principles  to  the 
case  of  Fig.  2,  and  replacing  F'  in  equa 
tion  5,  by  its  value  111,  we  shall  have  for 
the  horizontal  thrust  of  the  water  on  the 
face  A  B  C  D  of  the  dam  the  equation 
P"  —  ^h*ly,  and  in  the  same  way  the 
vertical  component  will  be  found  to  be 
equal  to  P'—  -J  a  h  I  y.  Now,  b  being  the 
breadth  of  the  dam,  and  ar  the  projec 
tion  of  the  slope  G  K,  and  y'  the  density 
of  the  masonry  composing  the  dam,  it  is 
evident  that  the  area  of  K  C  E  G  will 


/T     a  +  af\ 
be  tcH  --  -  —  I  fi  ;   the   cubic   contents 


lily'.     The  whole  vertical  pressure  on 
the  base  will  therefore  be  equal  to  this 


18 


weight  plus  the  vertical  pressure  of  the 
water,  or 


We  have  seen,  however,  that  the  force 
which  tends  to  counteract  the  push  of 
the  water,  and  on  which  the  stability  as 
to  slipping  must  therefore  depend,  is 
equal  to  this  weight  of  the  dam  increas 
ed  by  the  friction  of  the  stones.  De 
noting  this  co-efficient  of  friction  by/, 
we  shall  then  have  for  the  force  to  push 
the  dam  forward  the  expression 


and  in  the  case  where  the  horizontal 
thrust  of  the  water  is  to  effect  the  dis 
placement 


or   dividing   each  member   through   by 
^  h  ly^  we  shall  have 


19 


In  order  therefore  that  the  dam  may  not 
be  pushed  away  by  the  water,  we  must 
have  one  of  the  two  following  conditions 
fulfilled ;  either 


For  safety,  we  may  further  assume 
that  the  base  of  the  dam  is  quite  per 
meable,  in  which  case  there  is  (on  the 
principle  that  a  pressure  in  one  direction 
produces  an  equal  pressure  in  the  oppo 
site  direction)  a  pressure  from  below  up 
wards  equal  to  (2  b  +  a  +  af)  Ihy,  equal 
the  weight  of  the  dam,  and  as  this  is,  of 
course,  to  be  subtracted  from  the  above, 
we  have  finally, 


(2  b  +  a  +  a')        -  -  l-a'    9. 

These   equations   are    applicable   not 
only  to  the  sliding  of  the  entire  dam  on 


20 

its  foundation,  but  also  to  any  particular 
layer  of  stone  at  any  point  in  the  dam. 
The  value  of  the  co-efficient  of  friction 
f  will  of  course  be  very  different  in  cases 
where  we  consider  the  stability  of  differ 
ent  parts  of  the  wall,  from  that  in  cases 
where  we  consider  the  dam  to  slide  on 
an  earthty  foundation.  In  the  former 
case,  it  is  that  of  masonry  on  masonry, 
in  the  latter,  that  of  masonry  on  earth, 
and  in  general  clay.  In  fact,  it  may  be 
restricted  almost  solely  to  clay,  because 
in  a  sandy,  porous  or  yielding  soil,  it 
is  better,  on  principles  of  economy,  not  to 
build  a  dam,  but  a  dyke.  For  masonry 
on  masonry,  or,  indeed,  bricks  on  bricks, 
we  may  with  safety  take  the  co-efficient  of 
friction  as  equal  to  .67  ;  for  masonry  on 
dry  clay  .51;  but  for  masonry  on  wetted 
clay  the  co-efficient  falls  to  .33. 

A  few  examples  may,  perhaps,  serve 
to  illustrate  the  above  remarks.  We 
shall  confine  ourselves  first  to  the  case 
of  rotation  about  one  of  the  joints,  as 
that  is  really  the  most  likely  one  to  arise 
in  practice  : 


21 

Let  Fig.  la  represent  the  profile  of  a 
FlG.U.          A  D 


dam,  constructed  say  of  brickwork 
weighing  112  pounds  per  cubic  foot. 
Let  the  thickness  on  top  be  10  feet,  and 
that  at  the  base  20  feet,  required  to  find 
the  perpendicular  height,  the  dam  must 
have  in  order  that,  when  the  water  stands 
at  the  brim,  the  wall  shall  be  just  on  the 
point  of  turning  about  the  point  B  under 
the  pressure  of  the  water.  Denote  by  h 
the  height  of  the  dam,  or  the  quantity 
we  are  in  search  of,  =  C  D.  Now,  by 


22 


equation  2,  the  thrust  of  the  water  on 

one  lineal  foot  of  surface  is  —  X62.5  Ibs., 

A2 
and  the  moment  of  this  thrust  is  —  X62.5 

h       A3 

Ibs.  X  -  or  —  X  62.5  Ibs.     The  pressure 
o         6 

of  one  foot  of  the  dam,  or  what  is  the 

AD  +  BC  z 
same  thing,  its  weight  is  --  h  X 

104-20 
112  Ibs.,  or  ———h  X  112  lbs.=  1680  h 

Ibs.,  and  the  moment  of  this  pressure 
with  reference  to  the  point  B  is  1680  h 
XBE.  Before  we  can  obtain  this  mo 
ment,  then,  we  must  find  the  value  of 
B  E,  and  this  is  found  as  follows  : 

It  is  evident  from  a  moment's  inspec 
tion  of  Fig.  la,  that  the  area  of 
ABCDxG<7  =  area  ABC  FxBH  + 

area  of  ABFxIB,  or 
denoting  A  D  by  a\  B  C  by  ~b\  D  C  by  c; 
and  G  g  by  d,  we   have   since   B  H  = 

2b—a  2  (b—a) 

--,andIB=.         -L. 


2  b—  a     c(b—a) 
~~ 
dividing  by  c 


23 


.          11—  a     (b—aY 

—      - 


Substituting  for  the  above  quantities 
their  values,  we  have  : 

c?=^-W  =  1JiI, 

The  moment  of  the  dam  therefore  is 
1680  h  X  -%*. 

•'•    |pX  62.5  Ibs.  =  1680  A  X  JR 

62.5  A3  _184800A 

6  9 

^2  —  A/197.12 
A  =  44.3982. 

Again,  preserving  the  same  dimensions, 
let  it  be  required  to  find  the  "  modulus 
of  stability"  of  a  masonry  dam  of  the 
profile,  shown  in  Fig.  1,  the  stone  weigh 
ing  200  pounds  per  cubic  foot.  Draw 
from  the  middle  of  the  top  A  D  to  the 
middle  of  the  base  B  C  the  line  R  Y,  and 
take  its  length  as  45  feet,  and  the  depth 
of  the  water  behind  the  dam,  44  feet. 


24 


Now,  by  geometrical  principles,  which  it 
is  not  worth  while  to  repeat  here,  we 
have  : 


10  +  10  _  245 
15       =    T5~ 

g  being  the  centre  of  gravity  of  the 
wall.  Again,  in  the  two  similar  triangles 
E  Y  S  and  g  V  T,  we  have  : 

RV:  VS;:V<7:  VT. 

The  value  of  N  g  we  have  just  found. 
VS  is  evidently  equal  toYC—  SO,  or 
10  —  5  =  5.  In  the  triangle  RYS,  we 
also  have  R  S'=R  V'-VS2,  or  RS3= 
(45)2-(5)2;  hence  RS=44.38.  Substi 
tuting  these  values  in  the  above  propor 
tion,  we  shall  have  : 


45  :  5;:-     :  VT 
15 


The  weight  or  pressure  of  the  wall 
acting  through  the  centre  of  gravity  g 
of  the  dam  is,  as  we  have  already  seen, 

20+10 
—  -  X  1  X  44.38  X  200=133140  Ibs., 


25 

and  that  of  the  water  44X1X^X62.5 
=  60500  Ibs.  If  now  we  denote  by  P 
the  "  centre  of  pressure  "  of  the  water, 
that  is  to  say,  that  point  where  a  single 
pressure  will  counterbalance  the  thrust 
of  the  water  against  the  entire  face  D  C 
of  the  dam,  then  P=C  P=^=14.6  feet. 
The  quantity  we  are  in  search  of,  the 
modulus  of  stability  of  the  wall  is  the 
ratio  of  T  B  to  T  O.  The  value  of  T  B 
we  have  already,  and  may  obtain  that  of 
T  O  from  the  proportion  that  the  press 
ure  of  the  dam  is  to  the  height  of  the 
centre  of  pressure  (P)  of  the  water 
above  the  base  of  the  dam  as  the  press 
ure  of  the  water  is  to  the  entire  pressure 
of  the  water  acting  on  its  centre  of  press 
ure  P.  Thus : 

133140  :  14.6!  '.60500  :  x 
aj=6.6=TV. 

Dividing  this  last  found  quantity  by  T  B, 
we  have  : 

rp  "TT         f»   /i 

i^-^=  --^  =  .53*1  =  modulus  of  stability. 
1  J3  llf 

In  a  well  built  structure,  this  quantity 


26 

should  never  be  less  than  .5,  hence,  as  in 
the  present  case,  the  modulus  is  somewhat 
above  this  value,  we  are  justified  in  re 
garding  the  dam  as  a  perfectly  stable 
structure,  when  the  water  is  not  over  44 
feet  in  depth. 

In  these  considerations,  we  have  taken 
no  account  of  the  resistance  offered  by 
the  adhesion  of  the  mortar.  Should 
this  be  taken  into  account  —  and  it  is  al 
ways  best  that  it  should  not  —  then  equa 
tion  9  will  require  to  be  modified  some 
what  as  follows  :  Let  H  equal  the  dis 
tance  of  the  centre  of  gravity  of  a  layer 
of  stones  below  the  top  of  the  dam. 
The  shove  of  the  water  tending  to  throw 
down  this  portion  of  the  dam  is,  as  we 


.       _  .  , 
have  just  seen,  -  ,  in  which  expression 

2i 

6  is  merely  a  short  notation  for  ly.  The 
forces  resisting  this  shove  are  the  friction 
of  the  two  layers  sliding  on  each  other, 
and  the  adhesion  of  the  masonry.  The 
first  is  proportional  to  the  weight  of  the 
masonry  above  the  stratum  in  question, 
and  the  second  or  adhesion  of  the  mason- 


27 

ry  is  proportional  to  the  thickness  of 
the  dam  at  this  point.  Representing  as 
before  the  co-efficient  of  friction  by/*,  by 
c  the  cohesion  of  the  mortar  per  unit  of 
surface,  by  s  the  area  of  the  upper  sur 
face  of  the  course  next  below,  and  by  5 
the  thickness  of  the  dam  at  this  section, 
we  shall  have  for  the  resistance  R  to 
sliding  : 


and  therefore,  in  order  to  insure  stabili 
ty,  we  must  have  : 


2 

or  clearing  of  fractions,  and  then  divid 
ing  by 


H2 


10 


Neglecting  the  adhesive  power  of    the 
mortar,  the  above  becomes  : 


ad' 


The  second  case  of  slipping,  or  that  of 
the  dam  on  its  foundation  will  rarely,  if 


28 

ever  arise,  when  the  dam  is  founded  on 
a  rock,  for  in  that  case  the  value  of  the 
co-efficient  of  friction  will  be  the  same 
for  the  horizontal  section  of  the  founda 
tion  as  for  any  section  of  the  masonry- 
It  is,  however,  very  likely  to  arise  when 
ever  circumstances  will  not  enable  us  to 
lay  the  foundation  on  bed  rock.  In  such 
cases  the  soil  will  almost  always  be  of 
an  argillaceous  nature,  for,  should  it 
prove  to  be  of  a  gravelly,  sandy  or  very 
permeable  character,  the  employment  of 
some  common  form  of  dyke  will  be  much 
preferable  to  the  construction  of  a  dam. 
We  may,  therefore,  reasonably  assume 
that  in  all  cases  where  the  foundation 
course  does  not  rest  on  a  rock  surface,  it 
will  be  laid  on  argillaceous  soil,  and  as 
this  will  readily  give,  under  the  action  of 
water,  a  slippery  slimy  surface,  we  must 
assume  a  co-efficient  of  friction  very 
much  less  than  that  used  for  masonry  on 
masonry.  With  this  point  kept  clearly 
in  view,  the  conditions  of  stability  will 
be  given  by  the  above  equations.  Yet 
there  are  one  or  two  other  considerations 


29 

that  must  not  be  overlooked.  Thus,  as 
the  stability  will  depend  in  large  meas 
ure  on  the  lateral  resistance  of  the  soil, 
it  is  not  sufficient  'to  be  sure  that  this 
resistance  is  large  enough  to  prevent  the 
sliding  of  the  wall,  but  is  also  necessary 
to  be  assured  that  at  any  point  of  the 
front  of  the  foundation  wall,  the  normal 
pressure  does  not  exce'ed  the  limit  R/  of 
which  the  soil  or  the  wall  is  susceptible. 
Again,  in  order  to  prevent  any  slipping 
likely  to  arise  from  the  lateral  compres 
sion  of  the  earth,  it  is  not  necessary  to 
interpose  any  packing  between  the  face 
of  the  wall  and  that  of  the  ditch,  and, 
finally,  that  in  all  cases  it  never  comes 
amiss  to  "  step  "  the  rock  or  the  earth  on 
which  the  foundation  course  rests,  a  mat 
ter  to  be  considered  more  in  detail  here 
after. 

SECOND   CONDITION    OF    STABILITY. 

To  return  now  to  the  second  condition 
of  stability,  namely,  that  in  no  point  of 
the  structure  may  the  material  employ 
ed,  or  the  soil  of  the  foundation,  be  re- 


30 

quired  to  bear  too  great  a  pressure.  For 
this  purpose  let  A  B  C  D  (Fig.  3)  repre- 
D  C 


Rc.3. 


sent  the  profile  of  a  dam.  Then  from 
the  principles  we  have  already  establish 
ed,  it  follows  that  any  section  of  this, 
equal  in  length  to  a  lineal  unit,  may  be 
considered  as  subject  to  the  action  of 
two  forces,  which  are,  respectively,  the 
vertical  component  P  of  the  resultant  of 
the  weight  of  the  structure  above  that 
unit,  and  the  horizontal  pressure  or 


31 

thrust  of  the  water,  and  the  horizontal 
component  F  of  the  thrust  of  the  water. 
In  the  section  A  B  C  D,  these  two  forces 
act  through  the  centre  of  gravity  G,  and 
produce  a  resultant  of  their  own  which 
cuts  the  A  B  at  E.  This  latter  resultant 
R  may  therefore  be  regarded  as  applied 
directly  to  the  point  E,  and  resolved  into 
two  components,  one  vertical  and  equal 
to  the  force  P,  and  one  horizontal  and 
equal  to  the  force  F.  The  horizontal 
force  tends  to  slide  the  wall  along  the 
base  AB.  This  we  have  considered. 
The  vertical  spread^  itself  over  the  base 
from  the  extremity  B,  which  is  nearest 
the  point  of  application  of  the  resultant, 
according  to  the  well  known  decreasing 
law.  Now,  in  all  works  on  mechanics, 
we  have  given  a  formula  which  applies 
to  a  homogenous  rectangle,  pressed  by  a 
force  acting  upon  one  of  the  symmetri 
cal  axis,  and  this  is  : 


•  •  (x) 

and  ,     1 


32 

Where  N  is  the  entire  load  or  pressure, 
and  D,  the  entire  area  of  the  surface 
pressed.  In  the  case  we  are  considering, 
the  quantity  1ST  in  equations  cc  and  /?,  is, 
of  course,  represented  by  P  the  vertical 
component.  iQ,  by  I,  if  by  this  letter  we 
designate  the  breadth  of  the  base  A  B, 
and  if  we  denote  the  distance  E  B  by  u, 
then  will  the  quantity  n  in  equations  oc 

and  ft  be  represented  by  — - — . 

Substituting  these  quantities,  we  shall 

have  : 

* 

,    3l—6u\     P 


-••- 


and 


:  12. 

3  u 


Equation  cc  is  applicable  in  all  cases 
where  ^<J,  and  therefore  equation  11  is 


33 


I  _  2  u 
applicable  when  —  -  —  <£;  that  is  when 

i 


Equation  /3  is  applicable  to  all  cases 
when  ^>J,  and  consequently  equation 

I  _  2  u 

12  to  all  cases  when  —  -  —  >  J,  or,  what 
l 


is  the  same  thing  when  ^^<  J  L  We  have 
seen  that  the  condition  of  stability  re 
quires  that  some  limit,  R',  should  be 
placed  on  the  pressure  each  superficial 
unit  is  expected  to  bear.  The  pressure 
at  the  point  B,  must  therefore  be  less  or 
never  greater  than  R',  and  we  shall  have 
according  as  u  is  greater  or  less  than  J  19 


and 

2P 

-_-or<R'  .....     14. 

3  u 

And  this  condition  is  to  be  fulfilled  for 
each  section  made  in  the  profile,  neglect 
ing  the  force  of  cohesion  of  the  mortar 
which  is  unfavorable  to  resistance. 

These  expressions  are   susceptible  of 
yet  further  modification,  if  we  introduce 


34 

into  the  calculation  the  maximum  height 
A  that  may  be  given  to  a  wall  with  ver 
tical  faces,  so  that  the  pressure  upon^the 
base  shall  not  exceed  the  limit  B/  of 
safety.  Indeed,  if  we  represent  the  den 
sity  of  the  masonry,  or  the  weight  per 
cubic  yard  by  d,  we  shall  have  R'  =  tf'A, 
and  the  above  equation  become  : 


r-7=or<A      .     .     15. 
o  I 

and 

-  =  or<d/A=§-5r=or<\  .  .  16. 
3  u  ud 

The  conditions  expressed  in  these 
equations  would  be  quite  sufficient  if  the 
water  was  always  up  to  the  top  of  the 
dam,  but  as  this  is  by  no  means  always 
the  case,  the  wall  must  be  capable,  even 
when  the  dam  is  quite  empty,  of  sup 
porting  its  own  weight  without  being 
subject  at  any  point  to  a  pressure  per 
unit  of  surface  exceeding  the  limit  d'A. 

In  this  case  the  resultant  of  all  the 


35 

forces  acting  on  the  wall  is  reduced  to 
the  weight  P',  and  denoting  by  K  A, 
the  distance  from  the  resultant  passing 
through  the  centre  of  gravity  of  Fig. 
(3)  to  the  nearest  extremity  A  of  the 
base,  by  u,  the  pressure  at  A, will  be  given 
according  to  circumstances  by  equations 
11  or  12,  and  the  stability  of  the  wall 
will  require  that  one  of  the  relations  ex 
pressed  in  equations  15  or  16  be  satisfied 
when  P'  is  substituted  for  P. 

The  next  step,  therefore,  is  to  determ 
ine  the  proper 

PEOFILE     FOE    A    DAM     HAVING     ONLY    ITS 
OWN    WEIGHT   TO    CAEEY. 

In  order  to  study  under  all  conditions, 
the  question  we  are  now  about  to  con 
sider,  it  is  perhaps  well  to  inquire,  in  the 
first  place,  what  form  it  is  most  conven 
ient  to  give  a  dam  having  only  its  own 
weight  to  carry,  in  order  that  each  point 
of  the  masonry  shall  not  be  subjected  to 
a  pressure  larger  than  the  limit  of  safety, 
and  then  to  determine  the  alterations 
which  economy  require  to  be  made  in 


36 

this  assumed  profile.  It  is  evident,  to 
begin  with,  that  when  the  height  of  the 
dam  is  such  that  it  does  not  go  over  the 
limit  A  (i.  e.  the  greatest  height  we  can 
give  to  a  vertical  wall,  without  the  press 
ure  on  the  base  becoming  larger  than 
H',  we  shall  be  quite  justified  in  giving 
the  dam  vertical  facings,  and  that,  in 
such  case,  the  load  for  each  unit  of  sur 
face  at  the  lower  part  will  be  somewhat 
less  than  d'A,  or  at  least,  never  greater. 
Again,  we  know  that  whenever  the  press 
ure  on  a  horizontal  surface  of  masonry 
is  larger  than  the  limit  of  safety,  we  may 
correct  this,  by  enlarging  the  area  of  the 
surface  pressed,  and  so  lessen  the  load 
on  each  superficial  unit.  And  these  are 
the  two  fundamental  principles  of  dam 
construction,  and  may  be  summed  up  in 
brief  as  follows  :  If  we  are  construct 
ing  a  dam  of  a  height  equal  to  or  less 
than  A,  and  having  only  its  own  weight  to 
support,  it  is  a  safe  practice  to  give  it 
vertical  facings  from  top  to  bottom.  If, 
however,  we  are  constructing  a  dam  of 
a  height  greater  than  A,  yet  having  only 


its  own  weight  to  support,  we  must  make 
the  faces  vertical  for  a  distance  from  the 
top  equal  to  A,  and  from  this  point  to 
the  base  slope  them  outward. 

A  dam  constructed  on  this  latter  prin 
ciple  would  give  a  profile  similar  to  that 
in  Fig.  4.  From  the  summit  A  B  to  the 
section  C  D,  the  pressure  per  superficial 


B 


F.c/K 


F.c.6. 


TH 


unit  is   nowhere  greater  than   tf'A,  and 


38 

therefore  from  A  to  C  the  face  is  verti 
cal,  but  below  C  D,  the  load  exceeds  the 
limit  and  increasing  at  each  section  to  the 
base,  and  hence  from  C  to  Y  the  face  is 
sloping.  And  just  here  we  are  met  by  the 
great  question  in  dam  construction  that 
of  profile.  Should  the  bulging  portion 
C  Y  Y'D,  be  bounded  by  right  lines  as 
in  Fig.  4,  should  it  be  stepped,  should  it 
be  curved,  and  if  so,  should  the  bound 
ing  curves  be  logarithmic  curves,  simple 
or  compound  ?  these  are  questions  we 
propose  to  consider. 

It  is  an  easy  matter  to  determine  the 
force  to  be  given  to  the  facing,  so  that 
the  condition  that  the  load  per  unit  of 
horizontal  surface  shall  never  go  over 
the  limit  tf'A,  shall  be  satisfied.  To  do 
this,  we  may  choose  arbitrarily  one  face 
and  then  determine  the  other,  but  if  we 
desire  to  use  the  minimum  of  material 
consistent  with  perfect  safety,  then  the 
wall  must  be  symmetrical  as  to  its  axis. 
In  such  a  case  as  that  illustrated  in  Fig. 
4 — that  of  a  high  masonry  dam,  whose 
height  is  greater  than  A — the  slopes 


39 

D  N  Y'  and  C  M  Y,  ought  to  satisfy  the 
'requirement  that,  if  in  any  section,  as 
M  1ST,  the  load  per  surface  unit  is  equal 
to  any  given  quantity,  the  pressure  will 
be  the  same  for  any  other  section  as 
m'  ri  ',  infinitely  near  to  it.  This  will  be 
fulfilled,  if  the  increase  given  to  the  base 
is  proportional  to  the  increase  of  press 
ure,  or  as  the  profile  is  to  be  made  sym 
metrical  to  the  axis  O  S,  if  the  increase 
of  the  half  surface  L  1ST  or  L  M  is  pro 
portional  to  the  increase  of  load  on  that 
half  surface.  If  we  denote  by  P  the 
pressure  on  L  1ST,  arising  from  the  weight 
of  the  structure  above,  and  a  the  surface 
of  this  section,  then,  it  is  evident,  the 
above  condition  will  be  expressed  by 


K.da.     .    .     .     17. 

In  which  K  is  a  constant  quantity, 
and  denotes  the  limit  of  pressure  on  the 
unit  of  surface  or  d'A.  Again,  by  #,  de 
note  the  dimensions  of  the  dam  in  the 
direction  perpendicular  to  the  section  we 
are  concerned  with,  and  by  x  the  length 
of  the  half  section  LN,  or,  to  express 


40 

it  mathematically,  the  abscissa  of  the 
curve  or  line  sought  (i.e.  DN  Y'),  and 
finally,  by  y,  the  distance  of  MN  from 
a  horizontal  line  taken  as  the  axis  of  x. 
Then  the  surface  a  will  equal  to  bx,  and 
consequently  an  increase  of  surface  as 
da  in  equation  17,  will  be  expressed  by 

da—  dbx 
and  moreover 

<JP=d'bxdy 

These  values  substituted  in  equation 
17  give  for  the  differential  equation  of 
the  curve, 

d'bx.dy='K.b.dx.     .     18. 
whence 

7          K  dx 

dy  =  w^ 

But  K  equals  the  limit  of  pressure  per 
unit  or  #'A,  and  this  value  replaced  for 
K,  we  shall  have 

7          #'A   dx  ,  dx 

dy=-r-irOTdy=*-z 

Integrating  this  between  the  proper  lim 
its,  we  shall  have 


-2/0  =  A  log  ...     19 


41 

Now,  from  this  equation  we  see  that, 
the  curve  being  referred  to  rectangular 
axes,  one  of  the  co-ordinates  is  equal  to 
the  logarithm  of  the  other,  and,  hence, 
the  curve  must  be  a  logarithmic  curve. 
Here  then  we  have  one  property  of  the 
curve  D  N  Y.  To  find  in  the  next  place 
the  origin  of  its  co-ordinates,  we  may 
make  in  the  foregoing  equations  #0  =  A, 
in  which  case  we  shall  have  : 


and  y°=0  •  2(X 


From  this  last  relation  it  is  quite  ap 
parent  that  the  origin  of  co-ordinates  is 
to  be  taken  at  a  point  where  the  value 
of  x  is  equal  to  that  of  A,  and  in  this 
point  the  tangent  to  the  curve  makes  an 
angle  of  45°  with  the  axis  of  x.  Re 
turning  now  to  equation  19,  let  us  replace 
y0  and  x0  by  their  respective  values, 
given  in  equation  20,  when  we  shall  have  : 

y=\  log.  x 

A  i 

or  passing  from  the  system  of  Napier  to 
the  common  system  of  logarithms, 


42 


#=2.302658509  A,  log. -T-    .     .     21. 

6    A 

This  curve,  when  constructed,  will  give 
the  form  of  the  facing  of  a  wall  of  in 
definite  height  for  which  the  pressure 
per  unit  of  surface  equals  the  limit  of 
pressure  K.  It  is  not  to  be  forgotten  in 
in  making  use  of  equation  21,  that  the 
direction  in  which  ?/'s  are  usually  esti 
mated  has  been  reversed  ;  in  other 
words,  y  when  positive  is  to  be  estimat 
ed  downwards,  and  when  negative  up 
wards,  or  in  the  direction  of  L  O.  Fig. 
5  represents  this  curve  constructed,  by 
assuming  the  pressure  limit  or  K  as 
132,000  Ibs.,  and  the  density  of  the 
masonry  as  double  that  of  water. 

In  such  a  profile,  as  Fig.  4  has,  the 
sloping  faces  below  C  D  being  bounded 
by  right  lines,  we  may  obtain  the  neces 
sary  breadth  of  the  base  Y  Y',  as  soon 
as  we  have  determined  the  height  and 


*  We  may  also  pass  from  the  Naperian  to  the  common 
system,  by  multiplying  the  Naperian  logarithm  by  the 
modulus  of  the  common  system,  which  is  0.434294.  Its 
logarithm  is  9.63TT84. 


43 


Fic.5. 


the  breadth  at  top.  Denote  by  1)  the 
breadth  at  top  A  B  ;  by  h  the  distance 
AC=A,  and  by  h'  the  distance  from  C 
to  the  base  Y  Y';  by  $'  the  density  of 
the  masonry,  and  by  x  the  quantity  we 
are  seeking  for,  or  the  base  YY'.  Then 
we  shall  have  : 


The  quantity  h  in  this  equation,  which 
is  merely  another  expression  for  the 
quantity  A,,  has  been  determined  by  a 


44 

number  of  investigators,  but  the  most 
reliable  results  are  those  obtained  by  the 
French  engineers,*  who,  in  the  construc 
tion  of  their  great  masonry  dams,  such 
as  Furens,  have  taken  the  limit  of  press 
ure  K  at  60,000  kilogrammes,  or  about 
132,000  Ibs.  per  square  metre,  and  K 
being  equal  to  tf'A,  and  tf'  being  equal 
to  2,000  kilogrommes,  A  becomes  equal 
to  30  metres.  As  we  shall  hereafter  see, 
however,  the  limit  of  pressure  varies  for 
the  outer  and  inner  face  of  the  dam. 

If,  again,  the  profile  adopted  be  such 
as  is  illustrated  in  Fig.  3,  that  is  to  say, 
if  the  faces  of  the  dam  slope  continu 
ously  from  the  top  to  the  bottom,  then 
the  thickness  or  breadth  of  the  base  will 
evidently  be  obtained  by  dividing  the 
product  of  the  height  of  the  wall  and 
its  thickness  on  top  by  the  difference  be 
tween  2  A  and  the  height.  For  dr  A  or 
the  limit  of  pressure  is  equal  to  the  area 
of  the  profile,  multiplied  by  the  density 
of  the  masonry  divided  by  the  thickness 
of  the  base.  In  the  figure,  the  area  is 

*  MM.  Delocre,  Sazilly  and  De  Graeff. 


45 


plainly  equal  to  half  the  sum  of  the  two 
parallel  sides  by  the  altitude,  and  denot 
ing  this  latter  by  H,  we  shall,  therefore,, 
have  : 


The  conditions  which  govern  the  con 
struction  of  such  a  dam,  and  the  height 
to  which  it  is  safe  to  build  it,  become 
from  this  equation  quite  apparent,  should 

we  make  H=  2  A,  then  x  would  equal  —  ,. 

and  the  base  of  the  wall  would  spread 
out  to  infinity.  Should  we,  upon  the 
other  hand,  make  H  greater  than  2  A, 
then  A  would  become  negative,  and 
hence  it  follows  that  the  greatest  height 
we  can  give  to  a  masonry  dam  with 
straight  sides  equally  inclined  from  the 
summit  and  not  go  over  the  limit  of  re 
sistance  for  masonry,  is  equal  to  twice 
that  of  a  wall  with  vertical  sides.  Yet, 
within  this  limit,  such  a  profile  for  a 
masonry  dam  of  any  height,  occasions  a 
gross  waste  of  material.  This  becomes 
strikingly  apparent,  if  we  compare  the 


46 

breadth  of  base  of  a  dam  constructed 
with  inclined  faces  from  top  to  bottom, 
with  that  of  a  dam  of  the  same  height, 
but  having  a  profile  such  as  that  of  Fig. 
4.  Suppose  each  dam  to  be  30  metres 
high  and  5  metres  thick  on  top;  required 
the  thickness  at  the  base.  For  the  first 
ease,  using  equation  23,  we  have  : 

30X5 


For  the  second  form  of  profile,  we  use 
equation  22,  and  have,  since  the  quantity 
h  equals  A,  the  same  value,  or  x  =  5 
metres. 

If  we  raise  the  dam  by  10  metres,  then 
equation  23 

B=*>^=10  metres. 

60  —  40 

and  by  equation  23 

30X5  +  10 
or  since 

0  O     i^/&-j-/&i 

:    *,  ,\  7 — 77^=®=?  metres. 


47 

If,  once  more,  we  add  ten  metres  to 
the  height,  then  equation  23 

x=25  metres, 
and  eq.  22      cc=10  metres. 

The  saving  thus  affected  when  the 
dams  are  of  great  height  becomes  simply 
enormous.  The  difference,  however,  be 
tween  the  profile  when  the  dam  below 
C  D  (Fig.  4)  is  bounded  by  right  lines, 
and  when  bounded  by  logarithmic  curves, 
such  as  shown  in  Fig.  6,  is  not  so  marked 
as  in  the  cases  just  considered,  yet  is 
considerable.  To  take  but  one  case  in 
illustration,  a  dam  of  a  profile  such  as 
Fig.  6  illustrates,  with  the  faces  below 
CD  bounded  by  curves,  would  require 
(equation  21)  a  breadth  of  base  equal  to 
9.739  metres,  the  height  and  thickness 
at  top  being  as  before,  50  and  5  metres 
respectively,  while,  as  we  have  just  seen, 
if  the  faces  below  C  D  were  right  lines, 
the  base  would  be  10  metres. 

Such,  in  brief ,  is  the  relative  merit  of 
these  three  forms  of  profile,  for  a  dam 
having  nearly  its  own  weight  to  support. 


48 

In  practice,  however,  such  a  dam  can,  of 
course,  never  exist,  and  it  thus  becomes 
necessary  to  take  into  consideration  the 
second  condition,  or  that  of  a  dam  sup 
porting  a  charge  of  water. 

PROFILE    FOR    A    DAM    RESISTING   THE 
PRESSURE    OF    WATER. 

And  here,  again,  we  are  to  throw  aside, 
at  first,  all  practical  considerations,  and 
determine  a  theoretical  profile  of  equal 
resistance,  one  in  every  part  of  which 
the  pressure  shall  not  be  greater  than  the 
limit  R'.  For  this  purpose  we  return  to 
the  two  equations,  deduced  some  time 
back,  which  express  the  conditions  of 
stability  for  a  dam  resisting  the  thrust 
of  water,  and  neglecting  the  signs  > 
and  <  and  the  values  corresponding  to 
them,  take  only  those  corresponding  to 
the  sign  of  =.  We  then  have  the  two 
following  equations  : 


and  *»=* 


49      , 

If  we  now  replace  the  quantities  w,  I 
and  P,  by  their  respective  values,  ex 
pressed  in  functions  of  the  height  of  the 
dam,  we  may  readily  deduce  two  equa 
tions  which,  on  examination,  will  show 
two  things. 

1°.  That  the  profile  offering  the  least 
thickness,  consistent  with  the  conditions 
of  stability,  is  one  in  which  the  side 
turned  towards  the  water,  has  a  vertical 
face,  and  the  side  turned  from  the  water, 
or  the  outer  face  of  the  wall,  a  concave 
face. 

2°.  That  as  the  height  increases,  the 
thickness  increases  less  rapidly,  so  that 
in  a  wall  constructed  with  a  vertical  face 
on  the  water  side  and  a  curved  face  on 
the  other  side,  and  so  planned  that  it 
shall  satisfy  the  conditions  of  stability 
as  to  its  base  will  present  an  excess  of 
strength  for  the  surplus  of  height. 

Fig.  7  is  the  profile  of  a  dam  of  this 
description.  It  will  be  observed,  more 
over,  that  in  this  form  of  profile  the 
thickness  of  the  wall  at  the  top  is  zero 
This,  of  course,  in  practice  is  never  ad- 


50 


missible,  inasmuch  as  it  presupposes  the 
water  to  be  at  all  times  in  a  perfectly 
quiescent  state,  and  thus  makes  no  al 
lowance  for  the  very  considerable  force 
of  the  waves  raised  by  the  wind.  It  is, 
therefore,  necessary, whatever  the  profile, 
to  give  the  dam  quite  a  thickness  at  the 
summit,  in  general,  about  fifteen  feet,  is 


51 

a  good  width,  as  it  thus  enables  us  to 
construct  a  footpath  and  roadway  on  the 
top  of  the  dam,  which  is  quite  a  conven 
ience. 

Before  we  consider  any  other  modifi 
cations,  it  may  be  well  to  determine  as 
nearly  as  possible  the  co-ordinates  of  the 
concave  curve  forming  the  outer  face. 
For  this  purpose  we  will  take  the  verti 
cal  face  A  B  as  the  axis  of  #,  and  for  the 
axis  of  y,  a  perpendicular  to  this  pass 
ing  through  the  point  A,  and  call  it  AD. 
Anywhere  on  the  curve  we  will  take  a 
point  C,  and  denote  its  co-ordinates  B  C 
•=y  and  C  e=x  ;  then  the  relation  exist 
ing  between  x  and  y  will  give  the  equa 
tion  of  the  curve.  Now,  as  we  have 
already  seen,  the  wall  is  subject  to  the 
action  of  two  forces,  the  weight  of  the 
-  dam  P,  which  acts  vertically  downwards 
through  the  centre  of  gravity  and  the 
horizontal  thrust  T  of  the  water.  These 
two  forces  produce  a  resultant  R,  which 
cuts  the  base  of  the  dam  in  this  case  at 
the  point  H.  This  resultant,  therefore, 
may  be  regarded  as  applied  directly  to 


52 

the  point  H  and  resolved  into  two  com 
ponents,  HP  and  H  0,  respectively,  par 
allel  to  O  P  and  O  T.  We  have  also  seen 
by  equation  5  that  the  horizontal  thrust 
of  the  water  is  equal  to 

F'Ay=JAfty     ...     26. 

Or  replacing  h  by  its  value,  and  ly  by  its 
value  #,  then  T,  or  the  horizontal  thrust 
of  the  water,  equals 

T-  dx*  07 

~2~ 

And  in  the  same  way 


Returning  now  to  equations  24  and  25, 
we  find  that  the  quantity  I  is  equal  to  y, 
and  that  we  have  therefore  to  determine 
the  value  of  u  in  functions  of  x  and  of 
y.  Now  u  equals  H  C  and  H  C  =  K  C 
-  K  H.  The  triangles  O  P  R  and  O  K  H, 
moreover,  being  equiangular  triangles 
are  similar,  and  have  their  like  sides 
proportional,  and 


53 
KH  :  PR;;  OK  :  OP 


or 

KH     OK 


PR~~OP 

or  to  express  the  equality  in  terms  of  T, 
x  and  P, 

KH-*  29 

~T~~3P 

Replacing  in  the  29th  equation  the 
values  of  T  and  P,  as  obtained  in  the 
27th  and  28th  equations,  we  have  : 

KH_  x  -KH- ^_ 

*«•„*,/*     ,   "  ~3<r 


Ok 

Or,  for  brevity,  representing  - ,  by  D, 

TTTT  Da;3 

KH=    -jar      ...   so. 

6  /    11  d  x 
</    « 


54 

This  gives  us  the  value  of  K  H  in  the 
expression 

u=KC-KH.     ...     31. 

But  KG  is  evidently  equal  to  y— B  K, 
in  which  B  K  is  the  distance  from  the 
centre  of  gravity  of  the  surface  ABC 
to  the  vertical  axis  of  x  or  A  B.  This 
distance  is  equal  to  the  sum  of  the  mo 
ments  of  the  areas  such  as  a  b  c  d,  or 


or  again, 

/y 
if  dx 


BK= 


y 
Hence  KC  =  y-BK=y- 


y 

2  /   y  dx 

*/  o' 

y 
y*dx 


/y 
y  dx 

0* 

2  yj    ydw—J  y*dx 


32. 


f  ydx 


55 

Substituting  in  equations  31,  the  values 
of  K  H  and  K  C  obtained  in  equations 
30  and  32,  we  have  : 

rv  f*   a  D^3 

2V    ydx-J    y  dx     --  — 

u  = ~~  6  /    y  dx 

/y  */  /, 

T/cZcc 

0 

Or,  reducing  to  a  common  denominator, 
and  subtracting, 

/y  py 

ydx—3J    y*dx—Dx* 

u  —  _        ° °_ t    33, 


Thus,  then,  we  have  the  value  of  u  in 
functions  of  x  and  y,  and  substituting 
this  value  for  u  in  equation  24,  and  re 
membering  that  1=  7/,  we  have  : 

4yP— 6 ^P—1 

~^?~~ 

dx—36  y  $' 


56 


-18  tf'/V  dx-Q  Da?  S'fy 
/»y 

J  y  $x 


24  $'yfy  if  dx—3Q  y  d'f\fdx 

o 


i 
—A 


+  6  Dec3  c^ 

0 

Dividing  both  members  of  the  last  equa 
tion  through  by  6  &yf  y  dx,  wes  hall 

0 

have,  after  bringing  all  terms  containing 
y  into  the  first  member, 

-  2  )/y  j;  tf  a  +  3  fy  y  dx  +  Dec3—  A/  =  0 

o  o 

34. 

By  making  the  proper  substitutions  in 
equation  25, 


57 

/y 
ydx 
0J 

{§yj  ydx  —  3y  y*dx  —  T>x3 
0                                       0 

>  =7 

/•J 

6  /    ydx 

^    o                                   J 

0  .,  fv     7 
26  J   ydx- 

18  8\yfyydK--W\fyy*fo--§  3'XT)x: 

'         0                                    '         0 

/2/ 
y  dx 


18  S!\ 

0 

Transposing,  after  dividing  each  member 
by  3  tf'A,  we  have  : 


—  6  l^yy  dx  +  3  A 

+  X'Dx3  .     .     .     35. 

But  here  a  new  difficulty  presents  it 
self,  for  no  sooner  do  we  attempt  to  in 
tegrate  equations  34  and  35,  than  we  see 


58 

it  is  quite  impossible  to  perform  the  in 
tegration  by  any  exact  method.  We 
may,  however,  obtain  an  approximately 
correct  solution  by  finding  the  value  of 
y  in  a  series  of  functions  x.  Treating 
equation  34  by  this  method  we  obtain, 
says  M.  Delocre,  for  y  the  value 

r  i    t    t    V    v 

y=ax  +  bx  +  cx  +  dx  +  ex+fx+<&c.  36. 
While  equation  35  gives  : 

37. 
y — a  x  +  b  x*  +  c  x3  +  d  x*  +  e  x"  +f  x%  +  &c. 

These  equations,  as  it  is  quite  apparent, 
are  of  no  earthly  value  for  practical  pur 
poses,  and  we  shall,  therefore,  drop  all 
further  consideration  of  them.  Indeed, 
if  it  were  possible  to  obtain  the  equations 
of  the  curve  AmC,  by  a' short  and  sim 
ple  process  of  integration,  a  moment's 
reflection  will  show  that  such  a  profile 
as  that  illustrated  in  Fig.  7  would  not  be 
suitable  for  practical  use.  For  this  pro 
file  has  been  calculated  on  the  hypothesis 
that  the  dam  is  always  to  support  a  head 
of  water  equal  to  its  height,  and  in  this 


59 

case  the  pressure  on  any  horizontal  sec 
tion  as  m  n  will,  it  is  quite  true,  not  ex 
ceed  the   limit  R.     But  as  it  happens 
that  the  dam  is  very  likely  to  be  at  times 
empty,  the  profile  must  be  such  that,  full 
or  empty,  the  pressure  on  any  section 
as  m  n  shall  not  be  greater  than  R.    We 
know  that  this  limit  will  not  be  exceeded 
for  the   face  of    the  wall  bounded   by 
A  m  C,  and  it  thus  remains  to  consider 
only  the  vertical  face  A  B.  On  reference 
to  the  calculations  we  have  made  rela 
tive  to  the  profile  of  walls  having  only 
their  own  weight  to  support,  it  becomes 
noticeable  that   the  limit  will  soon   be 
passed  if    the   wall   is    slightly  raised. 
Supposing  this  limit  to  be  reached  at  the 
point  n^  we  are  forced  for  the  sake  of 
stability  to  depart  from  the  vertical  be 
low  this  point,  to  give  the  water  face  a 
swelling   or  bulging   surface,  and  -thus 
adopt  a  profile  similar  to  that  illustrated 
in  Fig.  8.     This  profile  is  supposed  to 
fulfill  the  conditions  that,  at  any  section 
as  de,  taken  below  mn^  the  pressure  at 
the  point  e,  the  dam  being  full,  will  be 


60 


less  than  or  equal  to  the  limit  R,  and  the 
dam  being  empty,  the  pressure  at  d  re- 


FIG,  8. 


DC  B 

suiting  from  the  weight  of  the  structure, 
will  also  be  less  than  or  equal  to  the 
same  limit  of  pressure,  R. 

This  last  modification,  moreover,  is  one 
of  no  small  importance,  as  it  enables  us 
to  correct  some  of  the  chief  errors  in 
which  the  theoretical  consideration  has 
unavoidably  led  us,  and  thus  to  approach 
nearer  to  the  end  in  view  ;  the  determi 
nation  of  a  profile  of  equal  resistance  suit- 


61 

able  to  practical  requirements.  If  the  two 
curves  mel>  and  ndT)  could  be  readily 
obtained  by  the  above  formula^he  profile 
of  Fig.  8  would  answer  almost  all  necess 
ary  conditionsas  testability  and  economy; 
but  they  cannot.  It  therefore  remains  to 
do  the  next  best  thing,  and  to  replace  the 
curved  surfaces,  by  polygonal  surfaces 
of  as  small  sides  as  possible — in  order 
that  they  may  approach  reasonably  near 
to  the  curves — and  then  determine  the 
equations  of  these  sides  of  the  polygons; 
or  to  adopt  a  similar  method  to  find  the 
equations  of  the  two  curves  in  question. 
This  we  shall  now  endeavor  to  do.  It 
is,  howrever,  to  be  remarked  that  there 
are  two  notable  instances  of  the  use  of 
the  form  of  profile,  shown  in  Fig.  8  ; 
that  of  the  dam  at  Furens,  and  that  con 
structed  on  the  Ban,  a  tributary  of  the 
Gier,  by  M.  Mongolfier.  Each  of  these 
we  shall  consider  later. 

As  this  form  of  profile,  therefore,  has 
been  illustrated,  and  its  economy,  dura 
bility  and  strength  fully  tested  in  the 
case  of  the  dam  at  Furens,  and  in 


62 

that  over  the  Ban,  we  shall  now  under 
take  its  investigation,  and  determine 
a  series  of  formulae  for  the  calculation 
of  the  logarithmic  curves  forming  the 
inner  and  outer  face  of  the  dam,  and, 
finally,  the  establishment  of  a  profile 
type  suitable  for  dams  of  various  heights. 
Our  investigation,  moreover,  is  to  be 
based  on  the  practical  experience  of 
MM.  Graeff  and  Mongolfier,  in  the  con 
struction  of  the  dams  of  Furens  and 
over  the  Ban,  and  the  brief  but  thorough 
report  of  Professor  Rankine  on  this  form 
of  profile,  to  many  parts  of  which  we 
are  greatly  indebted. 

In  the  first  place,  as  to  the  limit  of 
pressure,  two  questions  naturally  present 
themselves:  first,  what  shall  be  the  great 
est  limit  of  pressure  we  may  with  safety 
assume  ?  and  secondly,  is  the  same  limit 
to  be  adopted  for  the  inner  as  for  the 
outer  face  of  the  structure  ?  As  regards 
the  first  question,  it  becomes  evident  at 
a  glance  that  the  limit  R',  to  which  any 
point  in  the  dam  may  be  subjected  with 
out  thereby  endangering  stability, will  de- 


63 

pend,  to  no  small  extent,  on  the  nature 
of  the  stone,  cement,  or  mortar  used. 
Yet  here,  as  in  other  cases  where  mason 
ry  is  used,  it  is  possible  to  assign  a  gen 
eral  limit,  based  upon  practical  experi 
ence,  which  should  not  in  any  case  be 
overstepped,  and  if  possible  rarely  equal 
ed.  In  the  two  dams  to  which  we  have 
above  alluded,  the  limit  of  the  pressure 
was  taken  at  6  kilogrammes  per  square 
centimetre,  or  60,000  kilogrammes  to  the 
square  metre,  or  taking  the  kilogramme 
as  equal  to  2.20485  pounds,  132.291  Ibs. 
per  square  metre,  which  in  turn  is  equal 
to  1.1954  square  yards.  In  Spain,  how 
ever,  and  indeed,  we  believe  in  some  in 
stances  in  France,  the  limit  of  pressure 
has  been  taken  so  high  as  14  kilogrammes 
per  centimetre,and  the  dam  found  to  stand 
well,  but  in  the  majority  of  cases  at  from 
6  k.  to  8.50k.,  generally  at  6  k.,  per  square 
centimetre.  We  may  express  this  press 
ure  in  another  form  much  more  familiar 
to  English  engineers,  and  take  as  the 
limit  of  pressure  for  each  square  foot  or 
square  yard,  a  column  of  masonry  hav- 


64 

ing  that  area  for  a  base  and  a  height  of 
160  feet.  This  is  also  based  on  experi 
ence,  as  it  is  well  known  that  good  rubble 
masonry  will,  when  laid  in  strong  hy 
draulic  cement,  bear  with  safety  the 
pressure  arising  from  the  weight  of  a 
column  160  feet  in  height.  Taking,, 
again,  the  density  of  masonry  as  double 
that  of  water,  this  pressure  would  be 
equaled  by  a  water  column  320  feet 
high,  or  a  pressure  per  square  foot  of 
20,000  pounds. 

The  next  question  as  to  whether  the 
limit  of  pressure  should  be  the  same, 
both  for  the  inner  and  outer  face  of  the 
dam,  seems  to  be  viewed  very  differently 
by  different  engineers,  and  to  admit  in 
practice  of  a  variety  of  solutions.  In 
the  dams  constructed  by  M.  Graeff  and 
M.  Mongolfier,  and  in  the  theoretical 
profiles  offered  by  M.  de  Sazilly  and  M. 
Delocre,  the  same  limit  of  pressure  was 
adopted  for  each  face,  and  the  discussion 
of  the  formulae  thus  much  simplified. 
Yet  there  seems  to  be  much  ground  for 
departing  from  this  observance  and  for 


65 

adopting  two  limits,  one  for  the  outer 
and  one  for  the  inner  face,  provided  that 
the  dam  has  such  a  logarithmic  curve  of 
profile  as  that  we  are  considering.  It  is 
evident  that  the  vertical  pressure  along 
these  two  faces  is,  at  different  times,  un 
equal  ;  that  when  the  water  is  of  great 
depth  behind  the  dam  the  outer  face  is 
more  severely  strained  than  the  inner, 
and  that  when  the  water  is  very  low,  and 
the  dam  has  little  more  than  its  own 
weight  to  resist,  directly  the  opposite  re 
sult  takes  place  and  the  severest  strain 
is  found  along  the  inner  face.  It  is  like 
wise  evident  that  the  pressure  at  any 
point  along  these  faces  must,  in  all  cases, 
be  of  necessity  in  the  direction  of  the 
tangent  to  the  surface  at  that  place.  If 
the  face  is  vertical,  the  quantity  we  de 
rive  by  the  usual  equations  is  the  true 
vertical  pressure,  or  rather  the  entire 
pressure.  But  when  the  surface  slopes 
off  from  the  vertical,  as  it  does  in  this 
case,  the  pressure  is  in  the  direction  of 
the  tangent,  is  inclined  to  the  vertical, 
and  the  quantity  which  the  formula  gives 


66 

us  is  not  the  entire  pressure,  but  only  its 
vertical  component.  The  whole  or  real 
pressure  of  course,  exceeds  this  vertical 
component,  by  a  ratio  which  grows 
greater  and  greater  as  we  pass  down  the 
face  of  the  dam  to  parts  where  the  bat 
ter,  or  slope  of  the  face,  departs  more 
and  more  largely  from  the  vertical.  But 
the  outer  face  has  a  very  much  greater 
batter  than  the  inner,  and  the  water  be 
ing  high,  is  subjected  to  a  much  greater 
strain,  so  that,  to  equalize  matters,  and 
not  allow  the  outer  face,  when  the  dam 
is  full,  to  suffer  a-  greater  strain  than 
the  inner  face  when  the  dam  is  empty,  it 
becc  mes  most  expedient  to  take  a  lower 
limit  for  the  vertical  pressure  at  the  out 
er  than  we  do  for  the  intensity  of  the 
vertical  pressure  at  the  inner  face. 

Adopting  this  view,  it  remains  to  fix 
these  two  limits  of  vertical  pressure.  On 
the  inner  face,  it  is  clear,  where  the  slope 
deviates  so  very  little  from  the  vertical 
that,  for  all  intents  and  purposes,  it  may 
be  safely  neglected,  we  may  take  that  we 
have  already  fixed  upon,  namely,  the 


67       . 

weight  of  a  column  of  masonry  160  feet 
high.  For  the  outer  face,  we  may  take 
a  pressure  whose  vertical  component  is 
represented  by  the  weight  of  a  masonry 
column  120  feet  high,  a  pressure  which 
has  been  deduced  from  the  practical  ex 
amples  of  M.  Graeff. 

The  next  matter  to  be  taken  into 
account  is  that  of  tension,  which  must, 
so  far  as  possible,  be  avoided  in  every 
portion  of  the  dam.  And  this  brings  us 
to  the  consideration  of  the  "  lines  of  re 
sistance,"  of  which  in  structures  subject 
ed  to  such  varying  pressure,  there  are  of 
necessity  two  ;  one  for  the  condition 
that  the  dam  or  reservoir  is  full  of  water, 
and  one  for  the  condition  that  it  is  empty. 
As  in  the  case  of  earth  retaining  walls 
and  buttresses,  these  are  lines  passing 
through  the  centre  of  gravity  of  each 
course  of  masonry,  and  may,  when  the 
faces  of  the  dam  are  rectilinear,  be  found 
by  any  of  the  formulas  used  for  such 
purposes.  They  bear,  therefore,  intimate 
relations  to  the  stability  of  the  dam,  the 
latter  decreasing  as  they  depart  from  the 


centre  of  thickness  and  near  the  faces. 
They  also  bear  relation  to  the  tension, 
and  in  order  that  the  latter  may  not  be 
come  appreciable  in  any  part  of  the 
structure,  they  must  not  deviate  at  any 
point  from  the  line  passing  through  the 
centres  of  thickness,  either  outward 
when  the  dam  is  full,  or  inward  when 
empty,  by  a  distance  greater  than  one- 
sixth  of  the  thickness  at  that  point. 

With  these  conditions  in  view,  we  now 
pass  to  the  consideration  of  the  profile. 

PROFILE  TYPE  FOR  DAMS  HAVING  CURVES 
FOR  BOUNDING  FACES. 

Let  Fig.  9  represent  the  profile  of  a 
dam  bound  by  logarithmic  curves,  the 
various  equations  relative  to  which  we 
wish  to  find.  Let  the  vertical  line  A  S 
represent  the  asymptote  of  the  curves, 
and  taking  the  origin  of  co-ordinates 
at  the  top  of  the  dam,  represent  by  x 
all  horizontal,  and  by  y  all  vertical 
measurements,  by  b  the  breadth  or 
thickness  of  the  dam  across  the  top, 


69 


18 

....  1-410 


~fr< 

k.i|/v_--\50 

ilA. 


— iW-\ 

_Lv.-&-\9° 


FIG.  9. 


70 

and  by  b'  the  breadth  at  any  other  place 
lower  down.  Also  let  s  represent  the 
sub-tangent  common  to  the  two  curves, 
and  represented  in  the  figure  by  that 
part  of  the  asymptote  contained  between 
F  and  G.  As  to  the  lines  of  resistance 
let  their  deviation  from  the  middle  of  the 
thickness  when  the  dam  is  full  and  empty 
be  expressed  by  the  letters  r  and  r'  re 
spectively,  and  by  R  and  R/  denote  the 
limits  of  pressure  ;  the  first  for  the  out 
er,  the  second  for  the  inner  face. 

Now,  adopting  Professor  Rankine's 
method  of  procedure,  it  becomes  evident 
that  if  the  thickness  across  the  top  be 
expressed  by  #,  then  the  thickness  at  any 
other  portion  of  the  dam  lower  down, 
and  at  a  distance  y  below  the  top,  will 
be  expressed  by  the  equation 

br  =  b.e-    •  38. 

s 

in  which  e  is  the  modulus  of  the  common 
system  of  logarithms,  or  0.434294.  To 
apply  this  equation  therefore  to  practice, 
it  is  necessary  to  know  the  value  of  the 


sub-tangent,  the  thickness  across  the  top 
and  the  vertical  distances  of  different 
points  on  the  face  of  the  dam  below  the 
axis  of  X.  These  latter  points  are,  of 
course,  assumed  at  random,  and  have  in 
the  present  case  been  taken  five  feet 
apart.  As  to  the  thickness  at  the  top  it 
has  been  taken  at  eighteen  feet.  In  the 
dams  already  alluded  to  (those  of  MM. 
Sazilly  and  Mongolfier)  with  the  height 
of  50  and  42  metres  respectively,  and  a 
limit  of  pressure  of  60,000  kilogrammes 
per  square  metre,  the  thickness  across 
the  top  is,  in  the  former,  five,  and  in 
latter,  five  and  seven-tenths  metres, 
which,  expressed  in  feet,  gives  for  the  one 
16.4  and  for  the  other  18.6  feet.  But  in 
this  instance  we  have  slightly  enlarged 
on  the  thicknesses  used  by  thflse  engi 
neers,  in  order  to  produce  a  profile  suit 
ed  for  a  dam  required  to  resist  not  only 
the  thrust  of  water,  but  also  that  of  ice 
when  carried  down  by  spring  freshets. 
The  determination  of  the  sub-tangent  s 
is  not  so  obvious,  but  may  be  found  by 


72 

nj 

giving  to  the  exponent  -  of  e  an  approxi- 

«s 

mate  value  of  ,  which  substituted  in 
the  formula  of  Prof.  Rankine,  gives  a 
corrected  value  of  V,  and  a  sub-tangent 
equal  to  80  feet. 

If,  then,  adopting  this  breadth  of  18 
feet  on  top,  we  desire  to  find  that  at  a 
point  thirty  feet  below,  we  may  write 
equation  : 

O  A 

#'  =  log.  £  +  0.434294X  —      .      39. 
80 

=  1.255273  +  0.162858  =  26.19  feet, 

which  is  to  be  measured  off  in  such  wise 
that  thirteen-fourteenths  of  it  shall  lie 
on  the  down  stream  or  outer  side  of  the 
asymptote,  and  the  remaining  one-four 
teenth  on  the  up  stream  or  inner  side. 
Taking  other  values  for  y  and  proceed 
ing  in  precisely  the  same  way,  we  thus 
obtain  any  desired  number  of  points 
through  which  must  pass  the  logarithmic 
curves  that  form  the  faces  of  the  dam. 
This  done  and  the  curve  drawn,  the  next 
step  is  to  determine  the  lines  of  resist- 


73 

ance  when  the  dam  is  full  and  when  it  is 
empty.  To  begin  with  the  latter  case, 
the  dam  being  empty,  the  deviation  of 
the  line  of  resistance  from  the  middle  of 
the  thickness  will  evidently  be  inward 
or  towards  the  up  stream  side  of  the 
dam.  This  deviation  we  have  expressed 
by  the  letter  r' ',  and  if  we  wish  to  find 
its  value  for  a  horizontal  section  of  the 
dam  taken  50  feet  below  the  top,  we  pro 
ceed  as  follows.  Let  z  denote  the  dis 
tance  hg  or  the  deviation  of  the  centre 
line  of  the  thickness  outward  from  the 
axis  A  S,  and  by  zf  the  deviation  of  the 
same  line  from  the  same  axis  at  the  top 
of  the  dam.  Referring  to  Fig.  9,  the 
distance  we  wish  to  find  is  evidently 
equal  to  g  h  minus  the  deviation  of  the 
centre  of  thickness  of  the  top  of  the 
dam  from  A  S,  divided  by  2,  or 

r'=^  40 

_  «          .          •          •          ^rV»« 


Because  the  dam  having  only  its  own 
weight  to  carry,  the  line  of  resistance 
must  cut  the  line  ghiu  a  point  vertically 


74 

below  the  centre  of  gravity  of  that  part 
of  the  structure  above  g  A. 

The  thickness  of  the  dam  where  y  is 
fifty  feet  is  found  from  equation  39  to 
be  33.63  feet ;  the  centre  of  thickness 
16.81,  and  the  value  of  z  or  the  devia 
tion  of  this  centre  from  the  axis  A  S  is 
14.41  feet.  That  of  zr  or  the  deviation 
at  the  summit  of  the  dam  is  7.72  feet, 
from  which  it  follows  that  (eq.  40)  r'  = 
3.35  feet.  It  is  in  this  way  that  the 
values  of  r',  given  below  in  Table  A, 
have  been  calculated. 

It  is  next  necessary  to  determine  an 
equation  from  which  to  find  the  values 
of  r,  or  the  amount  by  which  the  line  of 
resistance  deviates  outward  from  the 
centre  of  thickness  when  the  dam  is  full. 
It  is  evident  this  deviation  will  depend 
upon  three  things,  the  moment  of  the 
horizontal  thrust  of  the  water,  above  the 
section  at  which  we  wish  to  find  r,  the 
weight  of  the  dam  above  this  same  sec 
tion,  and  the  amount  by  which  the  line 
of  resistance  is  moved  inward  when  the 
dam  has  only  its  own  weight  to  carry,  so 


75 

that  if  we  divide  the  moment  of  the 
thrust  by  the  weight,  and  subtract  the 
quantity  r',  we  shall  at  once  have  the 
value  of  r.  The  thrust  of  the  water 
above  any  horizontal  section  of  the  dam 
is,  as  we  have  already  seen  by  equation 

V2 
2,  —  X  62.5   Ibs.,  and   the   moment   is, 

2t 

therefore,  ^-X  62.5  X-=—  62.5  Ibs.,  or, 
2  36 

what  is  the  same  thing,  if  we  express  by 
w  the  ratio  in  which  the  masonry  is 
heavier  than  the  water,  and  take,  as  is 
usual,  this  ratio  as  2,  we  shall  have  for 
the  moment  (expressed  by  m)  of  the 
horizontal  thrust  of  the  water, 

V3      Vs 
m—-^- ¥-  41 

Qw     12 

The  weight  of  any  lineal  unit  of  the 
dam  above  the  section  may  be  found 
most  simply  by  the  calculus.  Thus  giv 
ing  to  y  and  b  the  same  signification  as 
before,  and  taking  the  weight  of  a  cubic 
unit  of  masonry  as  the  unit  of  weight, 


7G 


the  weight  of  each  unit  of  length  of  the 
wall  above  the  section  is  expressed  by 
nr      s*y 

J  v  dy  ...   42. 

0 

Integrating  this  between  the  limits  y 

and  o,  and  remembering  that  V=be^ 

$ 
we  have  : 


=  *  (*'-&)  .  .  .  43. 

For  r,  therefore,  we  have  : 


w 

y* 


This  equation  gives  for  the  value  of  r 
at  the  distance  fifty  feet  below  the  top, 
the  quantity  5.18  feet,  which,  as  it  falls 
below  one-sixth  of  the  thickness  at  this 
point,  we  are  justified  in  considering  the 


77 

deviation  as  not  too  great  to  be  perfect 
ly  consistent  with  stability. 

But,  to  make  assurance  doubly  sure, 
we  may  apply  a  final  test  as  to  stability, 
by  calculating  the  amount  of  vertical 
pressure  at  various  points  along  both  the 
inner  and  outer  faces,  and  comparing 
the  results  with  the  limit  of  pressure, 
which,  it  will  be  remembered,  has  been 
fixed  for  the  inner  face  at  weight  of  a 
column  of  masonry  160  feet  in  height, 
and  for  the  outer  face  at  that  of  a  col 
umn  120  feet  high.  This  matter  we  have 
already  considered  at  length,  and  have 
deduced  two  equations,  13  and  14,  which 
as  they  are  perfectly  suited  to  the  present 
case,  we  shall  not  delay  to  deduce  others, 
but  alter  them  to  suit  the  notation  of  Fig. 
9.  Thus  altered  they  are,  calling  p  and 
p'  the  pressures  at  the  outer  and  inner 
face  respectively,  and  P  and  P'  the  lim 
it  at  these  same  faces  — 


and  ^    45. 


78 

While  for  pr  we  have  two  others  precise 
ly  similar,  with  the  exception  that  P  in 
equation  45  is  changed  to  P'.  It  may, 
perhaps,  be  well  to  again  remark  that 
the  first  or  second  value  of  p  in  equtaion 
45  is  to  be  used  according  as  the  value 
of  it  is  greater  or  less  than  one-third  of 
the  thickness,  and  that  in  all  such  pro 
files  as  that  of  Fig.  9,  the  quantity  u  de 
notes  the  distance  from  the  outer  face 
to  the  line  of  resistance  when  the  dam 
supports  a  charge  of  water,  and  from  the 
inner  face  to  the  line  of  resistance  when 
the  dam  or  reservoir  is  empty.  To  illus 
trate  by  one  example,  let  it  be  required 
to  find  the  vertical  pressure  at  the  point 
C,  on  the  outer  face  of  the  dam  (Fig.  9), 
situated  fifty  feet  below  the  top.  By  re 
ferring  to  Table  A,  we  see  that  b'  is  equal 
to  33.63  feet,  that  the  outward  deviation 
of  the  line  of  resistance  is  4.98  feet,  and 
that  u  must  therefore  be  11.83  feet.  The 
quantity  W=s  (£'  —  £)  is  1250.4.  Since 

u  is  here  greater  than  —  =  11.21,  we  use 

o 

the  first  of  equation  45,  and,  making 
the  substitution  of  values,  we  have  : 


79 

=2  (2 
\ 


33.637  33.63 
Thus  showing  that  the  pressure  is  but  a 
little  more  than  half  the  limiting  press 
ure.  Precisely  the  same  operation  re 
peated,  with  u  equal  to  13.46  feet,  will 
give  the  amount  of  vertical  pressure  at 
the  inner  face  at  a  point  fifty  feet  below 
the  top,  the  dam  supporting  only  its  own 
weight.  This  pressure  is  thus  found  to 
be  equal  to  a  column  of  masonry  59.4 
feet  in  height. 

The  area  of  the  entire  profile  or  of 
any  portion  of  it,  included  between  two 
horizontal  sections,  may  be  found  by  tak 
ing  the  difference  between  the  thickness 
of  the  dam  at  these  two  sections,  and 
multiplying  the  difference  by  the  sub- 
tangent.  For  it  is  evident  from  the 
figure  that,  if  b  equals  the  thickness  of 
a  point  y  feet  from  the  top,  then  this 
thickness  multiplied  by  the  differential 
of  the  height  and  integrated  between 
the  limits  y  and  zero,  is  the  area,  and 

this  expression   /    bf  dy  when  integrat- 


80 


ed,  remembering  that  b'  is  equal  to  be- 

y  v        ^ 

gives  s  be  -— s b,  or  replacing  be  -  by  Z>', 
s  s 

the  expression  for  the  area  becomes 
s  (bf  —  b).  In  the  notation  we  have  used 
b  means  the  thickness  of  the  dam  across 
the  top,  but  in  calculating  the  area  of 
any  portion  of  the  profile  not  bounded 
by  the  top  thickness,  the  quantity  b  is  to 
be  understood  to  mean  the  smaller  of 
the  two  thicknesses  which  bound  the 
area.  That  is  to  say,  if  we  wish  to  find 
the  area  of  that  portion  of  the  profile 
included  between  horizontal  sections 
taken  at  thirty  and  eighty  feet  below 
the  top,  b  represents  the  thickness  at 
the  former  section,  and  we  have  80  (48.93 
—  26.19)  =  1819.2  square  feet.  Having 
the  area,  the  solid  contents  and  weight 
for  any  length  of  the  dam  are  of  course 
readily  found.  The  areas  for  sixteen  dif 
ferent  sections  of  the  profile,  each  hav 
ing  the  top  of  the  dam  for  one  side,  have 
been  calculated  in  this  way,  and  will  be 
found  entered  in  the  last  column  of 
Table  A.  The  first  column  of  this  table 


81 


gives  the  distances  in  feet  of  the  sec 
tions  estimated  from  the  top  downwards, 
the  second  the  thickness  of  the  dam  at 
these  sections,  the  third  the  deviation  of 
the  line  of  resistance  outward  when  the 
reservoir  is  full,  the  fourth  the  deviation 
inward  when  empty,  and  the  last  the 
areas. 

TABLE  A. 


r' 

r 

Area 
sq.  feet. 

0 

18.00 

0 

0 

o 

10 

20.40 

.51 

.18 

192.00 

20 

23.10 

1.09 

.54 

408.00 

30 

26.19 

1.75 

1.68 

655.20 

40 

29.68 

2.52 

3.18 

934.40 

50 

33.68 

3.35 

5.18 

1254.40 

60 

38.10 

4.53 

6.66 

1608.00 

70 

43.17 

5.39 

8.79 

2013.60 

80 

48.93 

6.62 

10.52 

2474.40 

90 

54.18 

7.75 

12.95 

2894.40 

100 

62.97 

9.63 

13.53 

3597.60 

110 

71.18 

11.39 

15.02 

4254.40 

120 

81.79 

13.62 

14.59 

5103.20 

130 

91.39 

15.72 

15.46 

5871.20 

140 

103.60 

18.34 

15.05 

6848.80 

150 

115.00 

20.78 

15.46 

7440.00 

160 

133.  GO 

24.64 

12.46 

9200.00 

82 

It  is  perhaps  unnecessary  to  call  at 
tention  to  the  fact,  that  this  form  of 
profile  has  been  calculated  with  a  view 
to  its  serving  as  a  profile  type  for  dams 
of  any  height,  great  or  small,  whose 
faces  are  logarithmic  curves.  For  a 
dam,  then,  of  which  the  height  is  thirty 
feet,  that  portion  of  Fig  9,  above  the 
line  marked  30,  is  the  proper*  profile  : 
for  one  eighty  feet  in  height,  that  por 
tion  above  the  line  marked  80,  and  so 
for  each  succeeding  section.  It  presents 
again  many  strong  points  not  found  in 
dams  of  the  usual  rectilinear  profile, 
which  are  especially  deserving  of  con 
sideration  when  damming  a  river  or 
valley  of  great  breath  and  depth.  Of 
these  not  the  least  is  its  economy  of 
material,  which,  as  we  shall  hereafter 
see,  is  very  great  as  compared  with  that 
of  stepped  or  sloping  profiles  ;  while  the 
curves  of  the  two  faces  are  so  gradual 
that  no  great  mechanical  difficulty  can 
arise  in  cutting  the  facings.  Another 
matter,  which,  in  the  dams  of  Furens  and 
the  Ban  was  not  taken  into  account, 


83 

that  of  tension,  has  here  been  considered 
and  the  profile  so  determined  that  when 
the  reservoir  is  full  the  tension  on  the 
outer  face  shall  not  at  any  point  be 
greater  than  it  is  on  the  inner  face  when 
empty. 

The  profile  of  the  Furens  dam  is  given 
in  Fig.  10,  and  that  constructed  on  the 


/1G./0, 


Ban,  a  tributary  of  the  Gier,  in  Fig.  11. 
The  former  has  a  height  of  fifty  metres 
with  a  breadth  on  top  of  5.70  metres, 


84 


A 


FIG.  11. 

and  a  limit  of  pressure  of  six  kilogrammes 
per  square  centimetre.  The  latter  has  a 
height  of  forty-two  metres,  a  thickness 
on  top  of  five  metres,  with  the  same 
limit  of  pressure  as  the  Furens  dam.  By 
a  comparison  however,  of  the  profile  of 
the  former  with  that  part  of  the  profile 
of  the  Furens  which  lies  above  the  limit 
A  B  we  see  that  the  thickness  has  been 
very  considerably  reduced,  while  if  we 
extend  the  profile  to  fifty  metres  and 


85 

then  compare  it  with  the  Furens,  we 
find  that  the  pressure  nowhere  exceeds 
8  kilogrammes  to  the  square  centimetre. 
To  return  now  to  the  modifications  of 
which  this  type  of  profile  is  susceptible. 

MODIFICATIONS  OF    THE    LOGARITHMIC 
PROFILE. 

On  a  moments  inspection  of  Fig.  8,  it 
is  readily  seen  that,  as  the  inner  curve 
does  not  anywhere  depart  very  far  from 
the  asymptot  AS,  the  first  and  simplest 
modification  of  this  curve  is  to  replace  it 
by  a  right  line  and  thus  make  the  inner 
face  vertical  from  top  to  bottom.  But 
the  outer  curve  if  treated  in  like  manner, 
and  replaced  by  a  right  line,  would  give 
us  a  form  of  profile  which,  though  it 
possessed  no  more  thickness  at  the  bot 
tom  than  was  absolutely  necessary  to 
withstand  the  vertical  pressure,  would 
at  every  other  point,  possess  a  thickness 
greatly  in  excess  of  the  requisite  amount, 
and  thus  occasion  a  prodigious  waste 
of  masonry.  We  must  therefore,  break 
this  continuous  slope  and  substitute  for 


86 

one  long  line  two  or  more  shorter  ones 
each  of  which  makes  a  different  angle 
with  the  vertical.  Limiting  our  atten 
tion  for  the  present  to  the  first  case,  and 
replacing  the  two  logarithmic  curves  in 
Fig.  9  by  lines, — the  inner  curve  by  one 
vertical,  and  the  outer  by  two  inclined — 
we  have  produced  for  us  a  profile  of  the 
form  illustrated  in  Figs.  12  and  13.  The 
question  that  first  presents  itself  in  the 
discussion  of  such  a  profile,  is  evidently 
how  far  down  the  outer  face  the  point  C 
is  to  be  taken.  It  comprises  indeed,  the 
entire  discussion.  Of  course,  it  is  a  great 
advantage,  so  far  as  the  saving  of  ma 
terial  is  concerned,  to  throw  this  point 
as  low  as  possible,  but  this  is  limited  by 
the  condition,  so  necessary  to  secure 
stability,  that  when  the  reservoir  is  full 
the  vertical  pressure  at  C  shall  not  be 
greater  than  the  limiting  quantity  R. 
Having  determined  the  thickness  across 
the  top,  which  preserving  our  previous 
notation,  we  will  call  b,  the  quantities  to 
be  determined  are  first,  the  vertical  dis 
tance  of  the  point  C  below  the  top,  and 


FIG.  12. 

second  the  thickness  of  the  dam  at  this 
point,  or  what  is  perhaps  more  easily 
obtained  the  excess  of  the  thickness  at  C 
over  the  thickness  at  the  top,  A  B.  The 


88 

distance,  AD  (Fig.  12)  we  will  call  y  ; 
78. 


FIG.  13. 

the  total  thickness  D  C  we  will  represent 
by  5',  and  express  the  excess  of  thickness 
by  v.  By  W,  denote  the  weight  of  the 
part  A  B  C  D  (Fig.  14),  and  by  F,  the 
horizontal  thrust  of  the  water  above  D. 
These  two  forces  act  through  the  centre 
of  gravity  O,  the  former  vertically 
downward  and  represented  in  Fig.  14  by 


the  line  O  P  ;  the  latter  horizontally 
and  represented  in  direction  and  inten 
sity  by  O  F.  These  two  produce  a  re 
sultant  which  cuts  the  base  at  V, 
and  this  point  may  therefore  be 
regarded  as  the  point  of  applica 
tion.  From  this  relation,  as  we  have 

seen,  result   two   equations   2  /2 ™\ 

P  2  P 

-—  or<R',  and  —  —  or<R',  which  are 

I  O     U  77 

to  be  used  according  as  u  is  >  -  or  <-. 

3  3* 

In  these  equations  P  =  W,  is,  accord 
ing  to  the  notation  of  Fig.  1 4,  expressed 


90 


by  (  -          -  1  y  #',  in  which   d'  is   the 
density  of  the  masonry  ;  l=V  =  ~b  +  v  and 


K  C  may  be  found  by  the  equation  ex 
pressing  the  relation  that  the  moment  of 
the  weight  of  A  B  C  D,  with  respect  to 
C,  is  equal  to  the  sum  of  the  moments 
of  the  two  parts  ABVD  and  BVC 
into  which  the  area  of  A  B  C  D  may  be 
divided.  The  moment  of  the  weight  of 
A  B  C  D,  with  respect  to  C,  is  evidently 

(9  £  _j_  -y  V 
—  —  1  y  df  multiplied  by 

KC  ;  that  of  ABVD  by  i^^ 

itfd' 
and  that  of  B  Y  C  by  J—  -  .     Hence,  the 

3 
relation  when  expressed,  becomes  : 


3 
rn-(fr  +  2«)»y*'. 


91 

~~6~b  +  3v 

To  find  KY,  we  have  from  the  two 
similar  triangles  O  K  Y  and  O  P  R  the 
proportion 

KY  :  KO;;PR  :  PC 

whence 

w     KOxPR 

K  V  — _ — or      .     .     48. 

since   P  R   is    equal    to   the   horizontal 
thrust,  which,  as  we  see  in  the  early  part 

if  d 
of  our  investigations,  is  equal  to    - —  ; 

m 

and  since  P  0  is  equal  to  the  vertical 

/2  Z?-}-v\ 
pressure  and  this  is  equal  to  I 1  yd 

we  have  finally  for  the  value  of  K  Y  : 

49. 
-    or 


which  latter  equation  is  found  by  substi- 

Ot 

tuting  for  -^7  the  letter  9.     These  values 

given  in  equations  49  and  47  when  re 
placed  in  the  expression 


02 


_ 


With  this  value  of  w  we  return  to  equa 
tions  24  and  25,  and,  substituting  it,  we 
obtain  : 

f        6 
2<{  2- 


X 


b'A 


/2 
V 


and 


These,  when  reduced  and  made  equal 
to  zero,  give  us  two  equations  containing 
two  unknown  qualities  : 

l>y—  hF  =  Q  .    51. 
*—  Q  b  /L  v 
0     .     .     52. 


93 
The  first  of  which  is  to  be  used  when 

u  >  --,  and  the  second  when  u  <  -.  Each 
3  3 

of  these  equations  express  the  relation 
that  when  the  reservoir  is  full  the  verti 
cal  pressure  at  the  point  C  (Fig.  14) 
shall  be  equal  to  the  limit  K.  But  we 
must  also  take  into  consideration  the 
inner  face,  and  find  an  equation  express 
ing  the  relation  that  the  reservoir  being 
empty,  the  pressure  at  Dr  shall  not  ex 
ceed  the  limit  R.  In  this  case,  the  face 
being  vertical,  the  pressure  of  the  water 
does  not  exist,  and  the  force  P,  or  the 
weight  of  this  portion  of  the  dam,  acts 
downwards  through  the  centre  of  gravi 
ty,  and 

w=DK=DC-OK 

2  v  (v  +  3  t>)  +  3  52 


_v(v  +  3  b)  +  3bz 


3 
With  this  value  of  u,  we  again  return 


94 

to   equations    24  and    25,   substitute  in 
each,  and  reducing,  have  : 

54. 

>/  y—  tf  A  -f  3  b  y  v—2  I  A  v  +  tfy—  A  £2  =  0 

55. 


By  combining  51  and  54,  or  52  and  55, 
we  may  readily  obtain  the  value  of  y 
and  v,  which  are  the  two  quantities  we 
wish  to  find.  It  is  moreover  to  be  re 
marked  that  A  in  the  above  equations  is 
found  by  dividing  the  limit  of  vertical 
pressure  at  0  and  D  by  the  ratio  in 
which  the  masonry  is  heavier  than 
water.  Thus  in  calculating  the  profile 
of  Fig.  12,  we  have  first  reduced  the 
limit  of  vertical  pressure  per  unit  of 
surface  from  pounds  to  kilogrammes, 
and  taking  the  density  of  water,  as  given 
in  the  French  tables,  as  1000  kilo 
grammes  and  the  density  of  masonry  as 
double  that  of  water  or  2000  kilo- 

R         60,000k 
grammes,  we  have  A  =  —  =  -^^ 

or  A  =  30.     We   thus   obtain    for   A,   a 


95 

very  simple  number,  whereas  had  we  re 
tained  the  pressure  as  expressed  in 
pounds,  we  would  have  had  a  much 
larger  one  to  handle.  In  Fig.  13  how 
ever,  in  order  to  produce  a  profile  of 
what  may  be  considered  as  a  type  of  the 
greatest  boldness  consistent  with  safety, 
we  have  taken  the  limit  of  vertical 
pressure  at  14  kilogrammes  per  square 
centimetre,  which  as  we  have  already 
stated  has  been  used  in  several  instances 
in  France  and  Spain.  This  increases  the 
value  of  A  to  70.  The  thickness  across 
the  top  is  in  each  case  the  same  as  in 
that  of  the  profile  illustrated  in  Fig.  9  ; 
namely,  eighteen  feet,  but  the  height  of 
that  in  Fig.  12  has  been  reduced  to 
ninety  feet.  The  height  AD  of  the 
upper  part  A  B  C  D  and  the  value  of  v 
corresponding  to  it  have  been  found  by 
combining  equations  51  and  54.  The 
lower  part,  by  the  same  equation,  by 
substituting  for  y  the  difference  between 
the  height  A  D  of  the  upper  part  and  the 
entire  height  of  the  dam. 

The  deviation  of  the  line  of  resistance 


96 


when  the  reservoir  is  full  may  also  be 
found  as  follows.  Let  A  B  C  D  in  Fig. 
15,  represent  either  the  upper  or  lower 


FIG.  15. 

part  of  the  dam  whose  profile  is  given 
in  Fig.  13,  and  let  it  be  desired  to  find 
the  amount  of  deviation  at  any  section 
as  E  F.  By  O  represent  the  centre  of. 
gravity  of  A  B  F  E,  then  will  O  B,  repre 
sent  the  resultant  of  the  two  forces  act 
ing  on  this  portion  of  the  dam,  and  the 
distance  we  wish  to  find  will  be  E  S. 
We  will  suppose  also,  in  order  to  cover 


97 

all  cases  that  the  water  stands  at  X. 
Also  let  A  B  =  5  ;  A  X  =  Z.  AE  =  y  ; 
E  S  =  a;  ;  E  W  =  A  ;  and  the  inclination 
of  the  sloping  side  B  C,  to  the  vertical 
be  denoted  by  cc  ;  by  #  the  density  of 
the  masonry  and  by  6'  that  of  the  water. 
Then  by  the  similar  triangles  O  P  R  and 
O  W  S,  we  have  : 


_ 
OW-QP 

Now  W  S  =  E  S  -  E  W  =  a  -  A  and 
OW  =  TE=JXE  because  the  centre 
of  pressure  (T)  of  a  rectangular  plane 
surface  sustaining  the  pressure  of  water, 
is  at  a  point  two-thirds  the  depth  of  its 
immersion.  Hence  T  E  =  J  (y  —  I).  PR 
or  the  horizontal  thrust  of  the  water 
on  XE  is,  as  we  know,  expressed  by 

—  iZ  --  L  .    an(j  the  pressure  O  P  by 

Zi 

(Z>  +  b')y  &         (2  b+y  tan.  oc  )  y  & 

x  —  h  _^br  (y  —  Q2  _  56. 
tan.  oc     & 


98 

r/ 

Then  replacing  —  by  6 

o 


7  _ 

/I  — 


an.c 
+  ty*  tan-  < 


2  by  +  y*  tan.  oc 
which,  added  to  equation  56,  gives  : 


58. 


This  value  of  &  is,  of  course,  to  be 
measured  off  from  the  vertical  side. 
When  the  water  stands  at  the  top  of  the 
dam,  the  value  of  /,  is  zero,  but  when 
the  reservoir  is  empty,  then  I,  is  equal  to 
the  entire  height  of  the  dam.  The 
simplest  way,  however,  to  find  the  devia 
tion,  is  by  means  of  Equation  50,  ob 
serving  that  the  value  of  £^,  when  found 
is  to  be  laid  off  from  the  outer  or  slop 
ing  face  of  the  dam  ;  and  corresponds  to 
the  distance  FS  in  Fig.  15. 

The  second  modification,  then,  of  the 
theoretical  profile  of  equal  resistance, 


consists  in  replacing  the  outer  curved 
face  by  a  broken  one  composed  of  two 
planes  inclined  at  different  angles  to  the 
horizon.  The  principles,  however,  which 
justify  us  in  the  use  of  such  a  modifica 
tion,  may  be  carried  still  further,  and 
the  inner  and  vertical  face  replaced  by 
one  almost  a  fac  simile  of  the  outer 
broken  one.  Indeed  the  only  essential  dif 
ference  between  them  lies  in  the  degree  of 
slope  which  we  give  to  their  two  plane 
surfaces.  On  the  one  side  both  are 
sloping  ;  on  the  other  that  portion  of 
the  face  from  the  summit  of  the  dam  to 
a  point  below,  (where  the  pressure  on 
each  unit  of  surface  equals  the  assumed 
limit  of  pressure,)  the  wall  is  vertical, 
and  from  here  to  the  base  slope  out 
ward.  This  latter  point  moreover,  must 
be  directly  opposite  that  point  on  the 
outer  face  at  which  the  two  sloping  lines 
of  the  profile  intersect.  Of  a  profile  thus 
constructed,  some  idea  may  be  had  from 
the  sixteenth  figure.  It  does  not  present 
any  merit  either  as  to  beauty,  strength, 
stability  or  economy  of  material  not 


100 

possessed  by  that  illustrated  in  Figs.  12 
and  13.  As  to  economy  indeed,  the 
amount  of  material  consumed  is  if  any 
thing  greater  in  former  than  in  the  two 
latter  forms  of  dams,  and  it  may  be 
justly  doubted  whether  the  additional 
stability  thus  obtained,  is  a  fair  recom 
pense  for  the  additional  outlay  for 
material  and  for  cutting  facing  stones 
for  a  third  sloping  face. 

As  to  the  mathematical  calculations  of 
such  a  profile  they  are  rather  lengthy  than 
difficult.  For  the  upper  portion  A  B  C  D, 
Fig.  16,  we  have  already  discussed  the 
principles  at  length,  and  obtained  in 
equations  51  to  55  the  necessary  formulae. 
The  value  of  A  B  or  b  is  of  course 
known,  as  also  that  of  AD  or  a'  which 
is  assumed,  and  is  not  to  be  greater 
than  A  or  the  greatest  height  we  can 
with  safety  give  to  a  wall  with  vertical 
faces.  That  of  the  lower  portion  C  D  E  F, 
may  also  be  conducted  on  the  principles 
previously  laid  down,  and  as  it  necessi 
tates  several  eliminations  of  somewhat 
startling  length  we  shall  consider  it 


101 


A        B 


merely  in  outline.  Knowing  the  total 
height  of  the  dam,  and  the  distance  A  D, 
we  of  course  know  D  G,  or  the  height  of 
that  portion  of  the  dam  C  D  E  F,  whose 
breadth  of  base  E  F,  we  wish  to  find. 
We  also  know  from  equations  51  and  54, 
the  breadth  D  C ,  and  projecting  this  on 
the  base  we  at  once  obtain  that  portion 


102 

of  it  between  GandH.  What  there  re 
mains  to  be  found  is  G  E,  and  H  F.  The 
former  of  these  unknown  quantities  we 
will  denote  by  y,  and  the  latter  by  z  ; 
the  breadth  E  F,  of  the  base  by  b,'  the 
part  G  H,  which  is  also  equal  to  C  D,  by 
#';  the  height  D  G,  of  the  lower  section 
of  the  dam  by  a,  and  that  of  the  upper 
section,  or  A  D,  by  af.  Returning  now 
to  the  equations  15  and  16,  which  are 
the  general  equations  of  stability  for  a 
dam  supporting  the  pressure  of  a  head 
of  water,  we  find  that  the  three  unknown 
quantities  for  which  we  wish  to  find 
values  in  term  of  the  known  quantities  we 
possess  are  it,  I,  and  p.  The  value  of  £, 
or  the  thickness  E  F,  of  the  base  is, 
when  expressed  in  terms  of  the  above 
notation. 

l=y+b+z 

While  P  is  of  course  the  area  of  the  ir 
regular  polygon  A  B  C  F  E  D  multiplied 
by  the  weight  per  unit  of  volume,  plus 
the  vertical  component  of  the  weight  of 
the  water  resting  on  the  sloping  face 


103 


DE.     The  area  of  ABCD  is  ( 

3'  a'.     That  of  C  D  E  F  is  y-?£-  +  Z^ 

4-  V  &  a.      The   vertical   thrust   of    the 

(2  a'  - 
— 2" 


o 
The  value  of  P,  therefore,  is  V  df  a  + 


y  #,  which  reduces  to  the  form 

P=  59. 


(2 


2 

Again,  to  find  the  value  of  u^  the  first 
step  is  to  construct  the  diagram  of 
forces,  as  illustrated  in  the  figure,  O  P 
representing  in  direction  and  intensity 
the  vertical  component  P,  or  the  weight 
of  the  dam  and  the  water,  and  O  F  the 
horizontal  component  or  the  outward 
thrust  of  the  water  behind  the  dam. 
Then  will  F  T  represent  u  which  is  clearly 
equal  to 

u=z  +  TLI—  IT      .     .     60. 


104 

But  by  the  two  similar  triangles  we 

O  "F1 

have,  as  before,  IT=OI  X  ~-p  or   since 


O  1=  -     -  and  O  F  (equation  2)  equals 


3 

'\  2 


('¥)'< 


yd] 

HI  is  to  be  obtained  in  precisely  the 
same  manner  as  K  C  was  obtained  from 
Fig.  14,  by  expressing  the  relation  that 
the  moment  of  weight  P  (which  includes, 
it  is  to  be  remembered,  that  of  the  dam  and 
that  of  the  water  pressing  on  the  inclin 
ed  face  D  E),  with  respect  to  the  point 
F  is  equal  to  the  sum  of  the  moments  of 
the  components  of  this  force.  Obtaining 
these  moments  in  the  same  manner  as 
we  obtained  those  for  the  equations  de 
duced  from  Fig.  14,  and  putting  them 
equal  to  the  expression  P  X  IF,  or  P  x 
(IH-f^),  we  have  after  reduction,  the 
equation 


105 


H= 

12  cc 


(1  2  cc  +  Vd)  +  6  a  z  4-  6ay  +  12  a'?/0  +  Qayd 

In  which  oc  is  a  short  expression  for  the 
area  of  A  BCD,  and  §  the  distance 
from  C  to  the  point  where  the  perpen 
dicular  of  the  centre  of  gravity  of 
A  B  C  D  cuts  C  D,  and  this  replaced  in 
equation  60,  gives  for  the  value  of  u 


12z(oc  +  b 

+  12oc/?  +  6£2a  +  2a;?/(2/  +  36')  +  3  (2  a' 
+  a)  (y  +  Zb^ye-Zatf-ie  (a' 


12(oc  +bf 

Eq.  61. 

The  quantities  P,  u  and  I,  being  thus 
obtained  in  terms  of  &',  y,  z,  a  and  a',  a 
substitution  in  equations  15  and  16,  will 
furnish  us  with  two  equations  of  great 
length,  from  which,  by  the  process  of 
elimination,  the  values  of  x  and  y  are 
readily  found. 

To  take  but  one  example  of  this  form 


106 

of  profile,  let  it  be  required  to  calculate 
the  dimensions  of  such  a  profile  for  a 
masonry  dam  one  hundred  and  seventy 
feet  in  height  and  eighteen  feet  broad  on 
top,  the  limit  of  pressure  being  taken  at 
132,000  pounds.  For  this  purpose  we 
have  to  determine  beforehand  the  height 
a!  of  the  part  A  B  C  D.  This,  in  the 
present  case,  is  taken  at  80  feet,  and  may 
in  all  cases  be  assumed  arbitrarily.  Now, 
since  the  dam  has  one  vertical  face,  we 
have  to  determine  but  one  quantity  v,  or 
the  difference  between  the  thickness  of 
the  dam  at  AB  and  that  at  CD,  and 
this  value  of  v  is  readily  obtained  from 
equation  51,  which,  modified  to  suit  the 
present  notation,  becomes 


Solving   this   with   reference    to   v,  we 
have 

b*a'  +  da"     72 

v  +2  o  v=  -  --  —  1} 
A 


And  replacing   the   quantities  by  their 


107 

values,remembering  that  A  equals  98.4  ft., 
and  9  (or  the  ratio  in  which  the  mason 
ry  is  heavier  than  water)  equals  £,  the 
result  finally  obtained  is, 

v  =  53.  52  —  18  or 

53.52  feet. 


With  this  value  of  b'  we  return  to  the 
equations  expressing  the  values  of  x  and 
y  as  deduced  from  equations  15  and  16, 
after  the  substitution  of  the  value  of  u 
given  in  equation  61,  and  find  that  the 
value  of  bff=x  +  b'  +  y  is  178.42  feet. 

Once  more,  we  may  carry  this  princi 
ple  one  step  further  and  produce  a  pro 
file  which  is  little  more  than  a  modifica 
tion  of  that  given  in  Fig.  16.  If,  for 
instance,  while  preserving  the  same 
height  of  structure,  we  divide  each  of 
the  three  sloping  faces  into  two  parts, 
and  give  to  each  part  thus  produced  a 
face  inclined  to  the  horizon,  we  shall 
then  have  a  profile  of  such  shape  as  that 
illustrated  in  the  seventeenth  figure. 

A  glance  at  this  is  sufficient  to  show  that 
it  is  in  reality  but  a  compound  of  the 


108 


A     B 


II  I 

two  preceding  profiles,  and  that  there 
fore  the  principles  to  be  observed  in  the 
calculation  of  its  parts  are  those  already 
discussed.  The  entire  profile  may  thus 
be  considered  as  divided  into  three 
pieces  ; — that  from  A  to  D,  in  which  the 
inner  face  is  vertical  throughout,  and  the 
outer  made  up  of  two  inclined  faces, 
constituting  a  profile  exactly  similar  in 
design  to  that  of  Fig.  12  :  that  from  D 


109 

to  F,  and  that  from  F  to  H,  in  each  of 
which  both  the  outer  and  inner  faces  are 
sloping.  The  first  part  is,  therefore,  to 
be  calculated  in  the  same  manner  as  we 
would  calculate  the  thickness  of  a  dam 
having  the  profile  of  Fig.  1 2,  and  each 
of  the  two  remaining  portions  by  the 
equations  deduced  from  Fig.  16.  To 
illustrate  this  by  a  case  in  point,  let  it  be 
required  to  find  the  thickness  at  various 
points  of  a  masonry  dam,  having  such  a 
profile  as  that  we  are  discussing,  its 
thickness  across  the  top  being  18  feet, 
and  the  total  height  170  feet.  The 
first  thing  that  claims  attention  is  the 
determination  of  the  vertical  distances 
between  the  points  B  and  C  ;  C  and  E  ; 
E  and  G  ;  and  finally  G  and  I.  These 
may,  of  course,  be  chosen  at  pleasure, 
just  as  we  may  select  the  number  of 
parts  that  each  face  is  to  be  composed 
of,  and  as  in  the  present  case  the  dam  is 
170  feet  high,  and  the  outer  face  divided 
into  four  parts,  we  will  for  convenience 
divide  the  dam  first  into  two  equal  parts, 
then  divide  the  lower  of  these  again  into 


110 

two  equal  parts,  and  the  upper  also  into 
two,  but  two  unequal  parts.  The  verti 
cal  distances  between  the  sections  will 
then  be,  beginning  at  the  bottom  and 
going  up  G I  =  42,5  feet;  E  G  =  42.5 
feet ;  C  E  =  45 ;  and  B  C  =  40  feet.  Had 
the  dam,  however,  been  one  hundred  and 
fifty,  or  one  hundred  and  eighty  feel 
high,  or  indeed  any  other  number,  then 
the  best  arrangement  would  again  have 
been,  to  make  the  second  vertical  dis 
tance — that  from  C  to  D  —  longer  than 
the  remaining  three,  so  that,  if  the  dam 
was  one  hundred  and  fifty  feet  high,  the 
best  arrangement  would  be  BC  —  30  ; 
CE  =  60;  and  E  G  and  GI  each  thirty 
feet  ;  if  the  height  had  been  one  hund 
red  and  eighty  feet,  then  B  C  =  40;  CE 
=  50  ;  and  the  others  each  forty-five 
feet.  Although  this  arrangement  may 
seem  to  be  somewhat  arbitrary,  it  is  in 
reality  based  upon  fixed  principles,which 
clearly  show  that  where  such  a  number 
of  divisions  and  such  a  profile  as  that 
used  in  the  present  instance  are  employ 
ed,  the  second  part  should  be  decidedly 


Ill 

longer  than  either  of  the  other  three. 
Those  portions,  moreover,  which  are 
bounded  on  both  sides  by  sloping  faces 
are  in  almost  all  cases  made  of  equal 
depth,  nor  does  there  seem  to  be  any 
reason  whatever  for  not  adhering  to  this 
method. 

With  these  distances  thus  determined, 
we  return  to  equations  51  and  54,  and 
from  the  first  of  these  find  the  value  of 
v,  as  was  done  for  equation  63,  and  sub 
stituting  for  ar  the  value  40,  and  for  b 
the  quantity  18  feet,  we  have 


98.4 

And,  consequently,  #'  =  £  +  ^  =  21.37  feet. 
To  find  the  value  of  b',  however,  it  is 
necessary  to  use  equations  51  and  54, 
from  which  by  the  common  method  of 
elimination  we  may  find  an  expression 

6y*  —  v*hy  —  3  byv  =  Q 

from  which  by  the  substitution  of  the 
proper  values  we  obtained  for  a  final  value 
of  //,  or  the  thickness  of  the  base  of  this 


112 

section,  £"=54.64  feet,  or  ^  =  33.27  feet. 
The  next  step  is  to  find  the  values  of  x 
and  ?/for  the  third  section.  As  this,  and 
also  the  last  section  have  both  faces  slop 
ing,  by  substituting  the  value  of  u  given 
in  equation  61,  in  equations  15  and  16,  and 
reducing  and  then  eliminating,  we  obtain 
two  expressions  for  x  and  y,  from  which 
we  derive  the  thickness  GF  =  100.36, 
and  by  a  similar  process  find  that  for 
I H  to  be  152.22  feet. 

It  is  thus  apparent,  that  as  there  is  al 
most  no  limit  to  the  number  of  sections 
into  which  a  dam  may,  on  this  principle,  be 
divided,  there  are  a  great  number  of 
different  forms  of  profile,  each  of  which, 
satisfy  the  conditions  of  stability,  but 
vary  somewhat  as  to  economy.  Theo 
retically  the  dam  whose  outer  face  con 
sists  of  the  greatest  number  of  these 
sloping  faces  is  the  most  economical, 
because  in  that  case  its  face  approaches 
nearest  to  the  logarithmic  curve  which 
bounds  the  theoretical  profile  of  equal 
resistance,  and  it  therefore  contains  very 
little  more  masonry  than  is  absolutely 


113 

necessary  to  insure  safety.  In  practice, 
however,  such  a  dam  would,  in  all  proba 
bility  prove  much  more  costly  than  one 
consisting  of  a  less  number  of  section, 
though  containing  more  masonry,  be 
cause  the  angle  of  inclination  of  the 
different  sections  of  the  outer  face 
changing  so  frequently  would  greatly 
increase  the  cost  of  cutting  the  facing 
stone.  To  avoid  the  mechanical  difficul 
ties  also  likely  to  arise  in  such  cases,  it  is 
sometimes  well  to  depart  altogether  from 
this  style  of  profile,  and  instead  of  slop 
ing  the  outer  and  inner  faces,  cut  them 
into  notches  or  steps. 

THE  STEPPED    PEOFILE. 

The  stepped  profile  has  been  reserved 
to  the  last  for  consideration,  because, 
while  it  is  a  natural  outgrowth  of  the 
preceding  modifications,  it  possesses 
many  merits  whose  importance  cannot 
be  fully  appreciated  till  a  comparison  is 
instituted  between  it  and  the  forms 
just  treated  of.  In  point  of  simplicity 
of  construction  for  instance,  it  would  be 


114 

difficult  to  find  any  design  of  profile 
that  can  surpass  it.  Wherever  the  faces 
of  the  dam  are  curved  as  in  Fig.  9,  or 
made  up  of  a  series  of  sloping  surfaces 
of  various  inclination  as  in  Figs.  12,  16 
and  17,  the  dimensions  of  every  facing 
stone  that  is  set  have  to  be  most  care 
fully  determined  beforehand  by  the 
rules  of  stereography,  and  this,  when  the 
dam  is  an  high  one  and  the  number  of 
stones  consequently  large,  is  of  itself  a 
work  of  no  small  difficulty.  In  the 
stepped  dam  however,  all  this  is  done 
away  with,  as  every  facing  stone,  (unless 
the  dam  is  curved)  possesses  only  a  ver 
tical  or,  if  it  happens  to  form  the  edge 
of  the  step,  a  vertical  and  horizontal 
face,  and  thus  requires  no  pattern  for  the 
stone  cutter.  A  further  advantage  to 
be  derived  from  it,  is,  that  it  enables  us 
to  approach  much  nearer  the  curved 
form  of  profile  than  we  can  in  any  other 
profile  type.  Indeed,  when  well  designed 
it  is  in  reality  nothing  but  the  logarith 
mic  curved  profile  cut  into  steps  or 
notches,  so  that  should  we  draw  a  con- 


115 

tinuous  line  through  the  upper  edges  of 
all  the  steps,  or  through  the  lower  edges 
of  their  vertical  faces,  this  line  would 
form  a  logarithmic  curve. 

Here,  as  in  the  calculation  of  the 
previous  profiles,  it  is  quite  allowable  to 
assume  arbitrarily  either  the  breadth  or 
height  of  the  step  and  from  this  one  de 
termine  the  other.  Yet  it  is  by  far  the 
best  plan  to  assume  the  vertical  height 
of  the  step  and  calculate  the  breadth. 
For,  it  must  be  apparent,  that  by  this 
method  of  procedure,  the  quantity  we 
calculate  is  really  the  abscissa  of  the 
curve,  which  we  lay  off  at  regular  inter 
vals  perpendicularly  to  the  vertical 
axis  of  the  dam,  and  in  this  way  we  are 
enabled  to  preserve  very  closely  the 
logarithmic  profile.  The  general  appear 
ance  of  the  dam  is,  moreover,  much  more 
pleasing  when  this  arrangement  is  ob 
served  than  when  we  assume  a  constant 
breadth  and  calculate  the  depth,  because 
the  breadth  of  the  steps  near  the  summit 
of  the  dam  is  then  very  narrow  and  in 
creases  gradually  as  they  approach  the 


116 

bottom,  and  the  departure  from  the  curve 
is  thus  scarcely  perceptible ;  but  when  the 
breadth  is  everywhere  the  same  and  the 
depth  varies,  the  whole  face  of  the  dam 
has  an  extremely  broken  appearance, 
which  is  anything  but  agreeable. 

In  this  profile,  as  in  all  the  others,  the 
inner  face  is  made  vertical  for  as  great  a 
distance  as  the  limit  of  pressure  will  al 
low,  and  from  that  point  down  it  is 
stepped.  The  outer  face  is  likewise 
made  vertical  for  a  distance  which  de 
pends  in  all  cases  on  the  thickness  across 
the  top,  being  as  a  general  thing  very 
nearly  twice  that  dimension.  In  the  de 
termination  of  the  following  formulae, 
the  depth  of  the  step  has  been  assumed 
as  the  same  throughout  the  entire  dam, 
and  the  breadth  has  been  taken  as  the 
unknown  quantity.  Fig.  18  then  rep 
resents  a  portion  of  the  profile  of  a  dam 
bound  by  a  curved  or  sloping  face, which 
we  wish  to  change  into  a  stepped  profile. 
ABDC  represents  this  section,  and  if 
H  F  be  taken  as  the  vertical  height  of 
the  step,  then  will  C  H  F  represent  the 


element  with  which  we  are  especially 
concerned,  and  its  base  CH  the  quantity 
we  are  in  search  of, — the  breadth  of  the 
step.  The  height  B  D  of  the  section  we 
will  denote  by  h\  and  the  density  of  the 
•masonry  by  d r;  and  the  greatest  thick 
ness  FT  or  HD  of  the  known  element 
ABTDHF  by  t\  from  which  three 
quantities  we  may  obtain  an  expression 
for  the  weight  P,  of  this  element,  which 
must  of  course  be  accurately  known,  in- 


118 

as  much  as  the  object  of  making  the  step 
at  this  point  being  to  lessen  the  amount 
of  vertical  pressure  on  each  superficial 
unit,  the  breadth  of  the  step  will  depend 
very  largely  on  the  weight  of  that  por 
tion  of  the  dam  which  is  above  it.  The 
weight  which  is  plainly  equal  to 


expressed  by  P,  while  that  of  the  ele 
ment  C  II  F  is  equal  to  -  ,  in  which 

a  is  the  height  of  the  step  F  H,  and  b 
the  breadth  C  H.  The  point  of  applica 
tion  of  the  thrust  of  the  water  is  T 
situated  at  two-thirds  the  depth  of  im 
mersion.  T'  and  T2  are  the  horizontal  and 
vertical  components  respectively.  Then 
will  P  represent  the  direction  of  the  re 
sultants  of  P  and  T2;  V  V  the  resultant 

of  P,  T2  and  the  weight  -   of  the  ele 

ment  C  H  F,  while  the  general  resultant 
of  all  the  forces  is  R.  Now,  in  this  case, 
as  in  the  previous  ones,  the  whole  solu 
tion  of  the  problem  depends  on  finding 


119 

the  value  of  C  R,  or  the  distance  from 
the  outer  edge  C  to  the  point  where  the 
resultant  cuts  the  base,  and  this  we  will 
express  as  heretofore  by  the  letter  u. 
Then  from  the  figure 

*4=CH  +  HY—  RV   ...     64. 

in  which  we  know  the  value  of  C  H=&, 
and  require  that  of  H  Y  and  R  Y.  But 

R  Y 

^r^f  is  equal  to  the  tangent  of  the  angle 

which  the  general  resultant  R  makes 
with  the  vertical,  or  calling  this  angle 
cc  then 

RY        T 

tan.  cc  = 


YY 

in  which  e  is  to  be  understood  to  express 

the  value  of  YY=^-FfT'      The   distance 
H  Y  may  be  found  from  the  theorem  of 


120 

moments,  by  expressing  the  relation  that 

d'tfa 


M- 


6 
d'Va 


>  +  *•» 

M  denoting  the  moment  of  P'  with  re 
spect  of  H.  As  to  C  H,  its  value  is  &, 
the  quantity  we  are  in  search  of.  Re 
placing  these  quantities  in  the  equation 
expressive  of  the  value  of  u,  we  have 

M  -  d'  W  a 


which,  reduced  to  a  common  denomina 
tor,  becomes 

65. 


u  = 


6P  +  3  6'  ba 


Having  thus  obtained  an  equation  for 
the  value  of  u,  the  next  step  is  to  find  by 
means  of  it  an  expression  for  b  the 


121 

breadth  of  the  step.  For  this  purpose 
draw  from  R,  the  point  at  which  the 
general  resultant  of  all  the  acting  forces 
outs  the  base,  a  perpendicular  R  N"  to  the 
resultant,  and  from  N  a  perpendicular  to 
the  base  C  D,  thus  forming  a  triangle 
R  N  O.  Then,  since  the  two  triangles 
R  Y  Y  and  R  1ST  O  have  their  bases  on 
the  same  right  line  C  D,  and  the  side  YR 
of  the  one  perpendicular  to  the  side  NR 
of  the  other,  and  the  sides  YY  and  NO 
parallel,  the  angles  at  Y  and  N  are  equal 
and  the  triangles  are  similar.  But  by 
the  relation  existing  between  the  sides  of 
such  similar  triangles,  we  have  the  pro 
portion 

N  O  :  R  Y  ; :  R  O  :  Y  Y. 

which  gives  for  N  O  the  equation 

RVXRO        T'/ 
~V~V^  ~        d'ab  •     •     •     66. 

~T 

in  which /is  the  distance  R  O.  But  we 
have  another  pair  of  similar  triangles 
which  gives  yet  another  value  for  N  O, 
which  must  be  deduced  and  made  equal 


122 

to  that  just  found.  These  triangles  are 
CO  1ST  and  C  H  F,  and  the  proportion 
derived  from  the  relation  of  their  sides 
is, 

NO:CO;:FH:HC  or 


_ 
H  C  b 

Equating  equations  66  and  67, 

COx£=/X—  ^, 

b  p     6  ab 

~§~ 

T' 

~ 


And  again,  since  if  four  quantities  be 
proportional  they  will  be  in  proportion 
by  composition  and  division 


> 

p+  -2-  •  b 

and  reducing, 


.123 

68. 


But  the  condition  of  stability  is  (equa 
tion  16)  expressed  by  the  relation 


/  = 


3  d'  A 


And  equating  these  values  given  in  equa 
tions  68  and  69, 


ua 


Substituting  for  u  its  equivalent  value  as 
given  in  equation  65,  and  dividing  both 
members  of  the  resulting  equation  by 
the  common  factor  2  P  +  d'  b  a,  there  re 
sults 

<T  A  «(6  b  Pf  +  3  tfd'a  +  6  M  -  6  T'e  -  d'Vd) 


Solving  this  with  respect  to  x  b*,  and  ex 
tracting  the  root, 


124 


3  A- 


9  T" 


4- 


(2  Tr 
6'  a 


2  a  A  -  a2- 


2  T' 


But  this  is  capable  of  being  yet  further 
reduced  by  dividing  through  by 


9  T' 

3h-^p--2a 
o  a 

2T' 

2  A-« — 

c> 


to  the  form 
5= 

P 


70. 


which  is  the  expression  for  the  breadth 


125 

of  the  step.  As  to  the  meaning  of  the 
letters  it  may  once  more  be  stated,  that 
P  is  the  weight  in  pounds  of  A  B  D  H  F, 
and  d'  the  density  of  the  masonry. 
The  vertical  height  (F  H),  which  we  de 
termine  to  give  the  step,  is  expressed 
by  a,  that  of  the  entire  dam  from  the 
top  to  the  base  of  the  step  by  A,  and  the 
moment  of  the  weight  P,  with  respect  to 
the  vertical  F  H  forming  the  rise  of  the 
step  by  M  ;  while  by  A,  we  mean,  as  in 
all  previous  formulae,  the  greatest  height 
to  which  we  can  raise  a  vertical  wall 
without  the  pressure  per  unit  of  surface 
on  the  base,  becoming  larger  than  the 
limit  R'  of  pressure  ;  and  by  $,  the  ex 
pression  — „  or  the  ratio  in  which  the 

density  of  the  masonry  exceeds  that  of 
water.  This  value  of  0,  is  safely  taken 
at  £.  As  to  the  height  to  be  given  to 
the  step,  this  is  of  course  to  be  assumed 
at  pleasure,  but  the  most  pleasing  effect 
is  produced  when  it  is  taken  at  six  or 
seven  feet,  for  then,  even  in  dams  of  one 
hundred  and  sixty  feet  in  height,  con- 


126 

structed  of  the  heaviest  stone,  the  breadth 
of  the  step  will  rarely  at  any  point  be 
materially  greater  than  the  rise.  The 
point  on  the  outer  face  at  which  the 
first  step  should  begin,  or  in  other  words 
the  distance  A  B,  in  Fig.  19,  is  deter- 
A 


127 

mined,  as  in  the  other  instances,  by  the 
relation  which  the  breadth  on  top  bears 
to  the  height.  If  the  thickness  t,  across 
the  summit  be  assumed  then 


4  Z4  2 


8  ^6**.*       6  A 

but  if  the  height  a  be  assumed  the  proper 
thickness  is  to  be  had  from  the  equation, 


=  a  4/     6  A^ 
- 


3  A  -  4 

When  that  point  on  the  inner  face  is 
reached,  at  which  it  becomes  necessary 
to  begin  stepping,  the  breadths  l>  and  &', 
of  the  outer  and  inner  steps  respectively, 
may  be  had  by  substituting  the  value  of 
u,  in  equations  15  and  16,  and  from  the 
two  resulting  equations,  finding  by 
elimination  two  expressions  for  b  and  V  . 
This  calculation  may,  however,  be  avoid 
ed,  and  considerable  expense  for  cutting 
facing  stone  saved,  by  making  the  inner 
face  vertical  from  top  to  bottom.  Indeed 
the  matter  of  expense  for  dressing  stone 
is,  perhaps,  the  most  serious  objection  to 


128 

the  stepped  profile,  as  it  is  necessary  to 
dress  both  faces  of  the  step. 

As  regards  the  use  of  the  formulae  for 
this  form  of  profile,  it  is  to  be  borne  in 
mind,  that  P  includes  the  weight  of  the 
water  as  well  as  the  weight  of  the 
masonry,  so  that  in  determining  the 
breadth  of  the  fourth  step,  the  weight 
of  the  three  columns  of  water  resting, 
one  on  the  first,  one  on  the  second  and 
one  on  the  third  step,  is  to  be  added  to 
the  pressure  of  the  masonry.  The  press 
ure  of  the  water  is  readily  obtained  from 
equation  1. 

The  principles  that  have  now  been  es 
tablished  in  connection  writh  the  four 
types  of  profiles  treated  of,  are  all  that 
are  required  to  calculate  the  parts  of  any 
profile  that  is  ever  likely  to  arise  in 
practice.  They  have,  moreover,  been 
determined  without  regard  to  the  length 
of  the  dam,  so  that  the  structure  will  be 
one  of  equal  resistance,  and  withstand 
the  thrust  of  the  water  solely  by  its 
own  weight.  There  is,  therefore,  no 
valid  reason  why  a  dam  constructed  with 


129 

a  profile  of  equal  resistance  should  be 
curved  into  the  form  of  an  arch,  and 
this  holds  good,  whether  it  be  high  or 
low,  whether  it  obstructs  a  broad  valley 
or  a  narrow  one.  The  only  thing  that 
can  be  accomplished  by  curving  a  dam, 
is  to  relieve  it  from  severe  strains,  by 
transmitting  as  large  a  part  of  the  thrust 
to  the  sides  of  the  valley,  but  where  the 
profile  is  such  that  the  dam  is  every 
where  equally  strong,  and  equally  capa 
ble  of  resisting  by  its  own  weight  the 
severest  strain  it  is  ever  subjected  to, 
there  is  surely  nothing  to  be  gained  by 
increasing  its  length  in  order  to  transmit 
this  thrust  laterally  to  the  sides  of  the 
valley.  It  is  true  that  in  deep  and  nar 
row  valleys,  some  saving  of  material  may 
be  affected  by  curving  the  dam,  which 
being  thus  relieved  from  a  goodly  por 
tion  of  the  thrust,  may  be  diminished  in 
thickness.  But  in  long  dams,  it  is  an 
open  question  whether  the  saving  thus 
affected  is  not  more  than  balanced  by 
the  increased  length. 

One  other  matter  which  deserves  the 


130 

most  careful  attention,    and   which   in 
deed  unless  it  is   carefully  attended  to 
will  render  the  very  best  profile  of  no 
account,  it  is  the  binding  of  the  stones, 
and   the  character  of  the   inner  filling. 
As  to  the  bond,  it  is  undoubtedly  the 
wisest  plan  if  the  dam  is  to  resist  a  great 
pressure,  to  avoid  laying  the  stones  in 
horizontal  courses  wherever  such  a  thing 
is  practicable,   and  to   place   binders  in 
every  possible  direction.     For  assuredly, 
if  it  is  necessary  for  the  stability  of  all 
walls  bearing  a  vertical  load,  that  there 
should   be   no  continuous   joints   in  the 
direction  of  the  pressure,  it  is  just  as 
important  that  a  dam   should  have  no 
continuous  horizontal  joints,  because  in 
the  case  of  such  structures  almost  every 
ounce  of   thrust  they  have  to   resist  is 
horizontal,   and   thus   exactly   coincides 
with  the  joints.     If  the  dam  is  curved, 
then  this   matter   of   broken   horizontal 
joints  is  not  of   such   vital  importance, 
because  no   layer  can   then   slide   until 
some  one  of  the  stones  has  been  crushed, 
yet  even  here  it  cannot  be  too  rigidly 


131 

adhered  to.  By  a  strange  inconsistency 
on  the  part  of  engineers,  we  often  see 
this  matter  both  regarded  and  disre 
garded  in  the  same  dam.  Many  struc 
tures  of  this  class  could  be  named,  in 
which  the  rock  foundation  is  stepped 
with  the  utmost  care  to  preclude  any 
possibility  of  sliding  where  sliding  is  of 
all  places  the  least  likely  to  occur,  while 
the  courses  from  the  foundation  to  the 
top  are  laid  with  the  most  perfect  kind 
of  horizontal  joints. 

The  filling  again  must  not  be  of  too 
different  a  character  from  the  facing. 
Where  masonry  consists  of  dressed  stone 
and  rubble  work,  the  amount  of  settling 
is  so  different  in  each  case  that  nothing 
like  a  bond  can  be  preserved.  The 
affect  of  such  settling,  we  constantly  see 
illustrated  in  the  most  striking  way  in 
canal  locks.  As  is  well  known  these  are 
generally  cut  stone  facings  with  rubble 
backing,  but  the  latter  settling  more 
than  the  former  become  detached  from 
the  facings,  when  the  water  penetrating 
between  the  two  kinds  of  masonry,  the 


132 

cut  stone  facings  fall  with  the  first  frost. 
A  good  filling  is  that  made  of  large 
rough  blocks  of  stone,  set  at  regular  in 
tervals  apart,  (the  distance  increasing  as 
the  top  is  approached)  and  the  spaces 
between  and  over  them  filled  in  with 
beton  of  the  first  quality,  a  method,  we 
believe,  lately  adopted  in  the  construction 
of  one  of  the  Croton  dams  in  this  state. 
But  perhaps  a  yet  better  one  is  to  replace 
the  beton  by  the  French  mixture  known 
as  beton  coigmt.  Both  of  these  fillings, 
however,  are  good,  as  when  well  rammed, 
they  form  a  close  connection  with  the 
facing  stones,  and  do  away  entirely  with 
joints  of  any  kind. 


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No   72.  — 

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-TOPOGRAPICAL  SURVEYING.  By  Geo.  J.  Specht, 
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Walling. 

SYMBOLIC  ALGEBRA  ;  or  The  Algebra  of  Algebraic 
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RECENT  PROGRESS  IN  DYNAMO- ELECTRIC  MA 
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MODERN  REPRODUCTIVE  GRAPHIC  PRO 
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STRUC"' 


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^' 
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.-Br 


cription  of  MinX  /*   i  * 

By  -Prof.  J.  C/  t  A  C   M 

^HET      J  V  V  /    I 
' 


S 


\  \ 

\ 


JE 
A&S  for 


jture,  Prop- 
kisser,  U.S.A. 

the  Gyroscope. 
,  )   G.  Barnard. 

LEVELING  :  Barometric,  Trigonometric  and  Spirit. 
By  Prof.  L  0.  Baker. 


Tl   UF7V. 


Q™ 
o.ilM 


P:      XV  _  BASIS  01 

I  -of.  T.  H,  HUXL^  ;,  _L.-b.  F.E.S.  "Wirl 
io  ~  /a  Profess  m  Y\Ie  College.  1 
P  3.:  *  "ers.  Jrice '•  -:3nts. 

-  • — '-1-L-i.E  cORTw  .  .  _  ..TCN  0^  V* 
T:> .x  iSICAL  FORO'E;  I  By  Prof.  Gteo* OT 
I.D.,  of  Yale  College.  86  p^.  Paper  Covci 

TIT—AS  BEGABP^  PEOTOPJ 
to  Prof.  Huxley's  Physical  Basif 
HUTCHISON  ^TIRLIN^,  F.7^.  ^,S.  pp.  72.  I 

TV.— ON  THE  F~    XX  HESIS  OF  F/{ 
Physical  and    Af#U,j    y$i\al.    By  Proi' 
COPE,     12mo.,  72  pp.   >apsr  Covers,     pr 

V.— SCIENTIFJC  ADDKESSES:— j. 
thoda  and  Tendvncif.s  of  Physical  /> 
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nation.    By  Prof.  JO^N  TYNDALL,  F.K. 
pp.  Paper  Covers.  Price  25  cents.    Flex.  ( 

NO.  VL—NATURAL  SELECTIOX  A 
TO  MAN.  By  ALFRED  RITSSELT,  WAI_ 
pamphlet  treats  (1;  of  the  Development 
Haces  under  the^aw  of  selection  ;  (2)  thelii 
ural  Selection  a?  applied"  to  man.  54  pp.  Pi 

NO.VlI.-SPECTKTJM  ANALYSIS, 
ture  i  b  T,  Prof  a.  Roscoe,  Muggins,  and  Loci 
88  pp.     Paper  Covers.     Pri 
-TI    -TFil   SulT.     A  sketch  of 
•  ""-V  •      "•/  jT^i^fi  as  regards  this  hodl 

0 :  »•.  - •"  "'         .          •"      recent  discoveries  and 
o     ar%  ,f.  f;  A,  YoiTNG,  Ph.I 

in'">rv.v  ^     ,...     Pap; 

N(. ,  ...\.       ?. ^H/  i 

A.  l-l.  '*  A-      -    7r      .,  n-  Bte>e^8  j 
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ihle  Cloth,  50  -      ' 

N( 
EAR. 


J  :-:c«  'Jo  ceif 

78TJJEIES  OF   THE   YOll 
^ror.  O.  N.  .Roop,  Columbia  Col] 


York,  f  One  -.  f  tha  "Tiost  interesting  Jectorea 
f  ver  deliverer!..  Orv.-i»«l  discoveries,  brilliaJ 
ments.  Beaut  •  luliy  illu«.  •' S8  pp.  Paper  Co vei 


